Mister Exam

Lang:

Other calculators

How to use it?
Least common denominator:
exp(x/2)/sqrt(1-exp(x))-exp(-x)/(2*sqrt(exp(-x)-1))
(((exp(1)-1)^3*log(e))/3)-((6*(exp(1)-1)-3*(exp(1)-1)^2+2*(exp(1)-1)^3)/18)
cot(x)^(atan(x)+1)*(log(cot(x))/(1+x^2)+(-1-cot(x)^2)*(atan(x)+1)/cot(x))
cot(x)^2*(-3-3*cot(x)^2)/tan(x)^2-cot(x)^3*(2+2*tan(x)^2)/tan(x)^3
Factor polynomial:
x^3+8*x^2-6*x-48
x^3-7*x^2+15*x
x^3+6*x*y+3*y^2-18*x-18*y
x^3+6*x^2*y+12*x*y^2+8*y^3
Factor squared:
-x^4+x^2-3
x^4-9*x^2+7
-x^4+9*x^2-5
-x^4+9*x^2+5
How do you in partial fractions?:
a-1/(3*a^2+6*a+3)-1/(2*a+2)
-2/(x-1)^2
(2tgx)/(1+tg^2x)
(2+sqrt(4+t)/sqrt(t+4))/(2-sqrt(t+4))
Identical expressions
cos(x-pi/ two)^(two)*cot(x+pi/ two)^(two)/(cos(x+pi/ two)+cot(x-pi/ two))*(cot(x-pi/ two)-cos(x+pi/ two))
co sinus of e of (x minus Pi divide by 2) to the power of (2) multiply by cotangent of (x plus Pi divide by 2) to the power of (2) divide by ( co sinus of e of (x plus Pi divide by 2) plus cotangent of (x minus Pi divide by 2)) multiply by ( cotangent of (x minus Pi divide by 2) minus co sinus of e of (x plus Pi divide by 2))
co sinus of e of (x minus Pi divide by two) to the power of (two) multiply by cotangent of (x plus Pi divide by two) to the power of (two) divide by ( co sinus of e of (x plus Pi divide by two) plus cotangent of (x minus Pi divide by two)) multiply by ( cotangent of (x minus Pi divide by two) minus co sinus of e of (x plus Pi divide by two))
cos(x-pi/2)(2)*cot(x+pi/2)(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cosx-pi/22*cotx+pi/22/cosx+pi/2+cotx-pi/2*cotx-pi/2-cosx+pi/2
cos(x-pi/2)^(2)cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))(cot(x-pi/2)-cos(x+pi/2))
cos(x-pi/2)(2)cot(x+pi/2)(2)/(cos(x+pi/2)+cot(x-pi/2))(cot(x-pi/2)-cos(x+pi/2))
cosx-pi/22cotx+pi/22/cosx+pi/2+cotx-pi/2cotx-pi/2-cosx+pi/2
cosx-pi/2^2cotx+pi/2^2/cosx+pi/2+cotx-pi/2cotx-pi/2-cosx+pi/2
cos(x-pi divide by 2)^(2)*cot(x+pi divide by 2)^(2) divide by (cos(x+pi divide by 2)+cot(x-pi divide by 2))*(cot(x-pi divide by 2)-cos(x+pi divide by 2))
Similar expressions
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x-pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)-cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cos(x+pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x+pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x-pi/2))
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)+cos(x+pi/2))
cos(x-pi/2)^(2)*cot(x-pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x+pi/2)-cos(x+pi/2))

Expression simplification/ Least common denominator/ cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))

Least common denominator cos(x-pi/2)^(2)cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))(cot(x-pi/2)-cos(x+pi/2))

The solution

You have entered [src]

   2/    pi\    2/    pi\                            
cos |x - --|*cot |x + --|                            
    \    2 /     \    2 / /   /    pi\      /    pi\\
-------------------------*|cot|x - --| - cos|x + --||
   /    pi\      /    pi\ \   \    2 /      \    2 //
cos|x + --| + cot|x - --|                            
   \    2 /      \    2 /

$$\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)}} \left(- \cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)}\right)$$

((cos(x - pi/2)^2*cot(x + pi/2)^2)/(cos(x + pi/2) + cot(x - pi/2)))*(cot(x - pi/2) - cos(x + pi/2))

General simplification [src]

   4    /       1   \
sin (x)*|-1 + ------|
        \     cos(x)/
---------------------
 (1 + cos(x))*cos(x)

$$\frac{\left(-1 + \frac{1}{\cos{\left(x \right)}}\right) \sin^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}$$

sin(x)^4*(-1 + 1/cos(x))/((1 + cos(x))*cos(x))

Rational denominator [src]

    2/    pi\    2    /   /    pi\         \ 
-cot |x + --|*sin (x)*|cos|x + --| + tan(x)| 
     \    2 /         \   \    2 /         / 
---------------------------------------------
             /    pi\      /    pi\          
          cos|x + --| + cot|x - --|          
             \    2 /      \    2 /

$$- \frac{\left(\cos{\left(x + \frac{\pi}{2} \right)} + \tan{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)}}$$

-cot(x + pi/2)^2*sin(x)^2*(cos(x + pi/2) + tan(x))/(cos(x + pi/2) + cot(x - pi/2))

Common denominator [src]

                                        5                         
       4         2       2         2*tan (x)           3          
- 2*tan (x) - sin (x)*tan (x) + --------------- + 2*tan (x)*sin(x)
                                sin(x) + tan(x)

$$- \sin^{2}{\left(x \right)} \tan^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \tan^{3}{\left(x \right)} - 2 \tan^{4}{\left(x \right)} + \frac{2 \tan^{5}{\left(x \right)}}{\sin{\left(x \right)} + \tan{\left(x \right)}}$$

-2*tan(x)^4 - sin(x)^2*tan(x)^2 + 2*tan(x)^5/(sin(x) + tan(x)) + 2*tan(x)^3*sin(x)

Numerical answer [src]

cos(x - pi/2)^2*cot(x + pi/2)^2*(-cos(x + pi/2) + cot(x - pi/2))/(cos(x + pi/2) + cot(x - pi/2))

cos(x - pi/2)^2*cot(x + pi/2)^2*(-cos(x + pi/2) + cot(x - pi/2))/(cos(x + pi/2) + cot(x - pi/2))

Combinatorics [src]

    2       2                       
-sin (x)*tan (x)*(-tan(x) + sin(x)) 
------------------------------------
          sin(x) + tan(x)

$$- \frac{\left(\sin{\left(x \right)} - \tan{\left(x \right)}\right) \sin^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}{\sin{\left(x \right)} + \tan{\left(x \right)}}$$

-sin(x)^2*tan(x)^2*(-tan(x) + sin(x))/(sin(x) + tan(x))

Powers [src]

                           2                                               
/   /    pi\      /pi    \\          /             /    pi\      /     pi\\
| I*|x - --|    I*|-- - x||          |           I*|x + --|    I*|-x - --||
|   \    2 /      \2     /|          |             \    2 /      \     2 /|
|e             e          |     2    |          e             e           |
|----------- + -----------| *tan (x)*|-tan(x) - ----------- - ------------|
\     2             2     /          \               2             2      /
---------------------------------------------------------------------------
                       /    pi\      /     pi\                             
                     I*|x + --|    I*|-x - --|                             
                       \    2 /      \     2 /                             
                    e             e                                        
                    ----------- + ------------ - tan(x)                    
                         2             2

$$\frac{\left(\frac{e^{i \left(- x + \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(x - \frac{\pi}{2}\right)}}{2}\right)^{2} \left(- \frac{e^{i \left(- x - \frac{\pi}{2}\right)}}{2} - \frac{e^{i \left(x + \frac{\pi}{2}\right)}}{2} - \tan{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\frac{e^{i \left(- x - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(x + \frac{\pi}{2}\right)}}{2} - \tan{\left(x \right)}}$$

   2       2                      
sin (x)*tan (x)*(-tan(x) + sin(x))
----------------------------------
         -sin(x) - tan(x)

$$\frac{\left(\sin{\left(x \right)} - \tan{\left(x \right)}\right) \sin^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}{- \sin{\left(x \right)} - \tan{\left(x \right)}}$$

sin(x)^2*tan(x)^2*(-tan(x) + sin(x))/(-sin(x) - tan(x))

Combining rational expressions [src]

   2/-pi + 2*x\    2/pi + 2*x\ /     /pi + 2*x\      /-pi + 2*x\\
cos |---------|*cot |--------|*|- cos|--------| + cot|---------||
    \    2    /     \   2    / \     \   2    /      \    2    //
-----------------------------------------------------------------
                     /pi + 2*x\      /-pi + 2*x\                 
                  cos|--------| + cot|---------|                 
                     \   2    /      \    2    /

$$\frac{\left(- \cos{\left(\frac{2 x + \pi}{2} \right)} + \cot{\left(\frac{2 x - \pi}{2} \right)}\right) \cos^{2}{\left(\frac{2 x - \pi}{2} \right)} \cot^{2}{\left(\frac{2 x + \pi}{2} \right)}}{\cos{\left(\frac{2 x + \pi}{2} \right)} + \cot{\left(\frac{2 x - \pi}{2} \right)}}$$

cos((-pi + 2*x)/2)^2*cot((pi + 2*x)/2)^2*(-cos((pi + 2*x)/2) + cot((-pi + 2*x)/2))/(cos((pi + 2*x)/2) + cot((-pi + 2*x)/2))

Assemble expression [src]

   2/    pi\    2/    pi\ /     /    pi\      /    pi\\
cos |x - --|*cot |x + --|*|- cos|x + --| + cot|x - --||
    \    2 /     \    2 / \     \    2 /      \    2 //
-------------------------------------------------------
                  /    pi\      /    pi\               
               cos|x + --| + cot|x - --|               
                  \    2 /      \    2 /

$$\frac{\left(- \cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)}\right) \cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)}}$$

cos(x - pi/2)^2*cot(x + pi/2)^2*(-cos(x + pi/2) + cot(x - pi/2))/(cos(x + pi/2) + cot(x - pi/2))

Expand expression [src]

   3       2          2       3    
sin (x)*tan (x)    sin (x)*tan (x) 
---------------- - ----------------
-sin(x) - tan(x)   -sin(x) - tan(x)

$$\frac{\sin^{3}{\left(x \right)} \tan^{2}{\left(x \right)}}{- \sin{\left(x \right)} - \tan{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} \tan^{3}{\left(x \right)}}{- \sin{\left(x \right)} - \tan{\left(x \right)}}$$

sin(x)^3*tan(x)^2/(-sin(x) - tan(x)) - sin(x)^2*tan(x)^3/(-sin(x) - tan(x))

Trigonometric part [src]

                                      2                                 
                     /           /x\ \          /                 /x\ \ 
                   2 |      2*tan|-| |          |            2*tan|-| | 
 /         1      \  |           \2/ |     2    |                 \2/ | 
-|1 - ------------| *|1 + -----------| *tan (x)*|-tan(x) + -----------| 
 |       2/x   pi\|  |           2/x\|          |                 2/x\| 
 |    tan |- + --||  |    1 + tan |-||          |          1 + tan |-|| 
 \        \2   4 //  \            \2//          \                  \2// 
------------------------------------------------------------------------
                          /       /x\          \                        
                          |  2*tan|-|          |                        
                          |       \2/          |                        
                        4*|----------- + tan(x)|                        
                          |       2/x\         |                        
                          |1 + tan |-|         |                        
                          \        \2/         /

$$- \frac{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(- \tan{\left(x \right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \tan^{2}{\left(x \right)}}{4 \left(\tan{\left(x \right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}$$

           /               3   \
      2/x\ |     3      sin (x)|
-4*tan |-|*|- tan (x) + -------|
       \2/ |               2   |
           \            cos (x)/
--------------------------------
             2                  
/       2/x\\                   
|1 + tan |-|| *(sin(x) + tan(x))
\        \2//

$$- \frac{4 \left(\frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \tan^{3}{\left(x \right)}\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

    4    /   sin(x)           \
 sin (x)*|----------- + sin(x)|
         \cos(x - pi)         /
-------------------------------
/             sin(x)  \    2   
|-sin(x) + -----------|*cos (x)
\          cos(x - pi)/

$$\frac{\left(\sin{\left(x \right)} + \frac{\sin{\left(x \right)}}{\cos{\left(x - \pi \right)}}\right) \sin^{4}{\left(x \right)}}{\left(- \sin{\left(x \right)} + \frac{\sin{\left(x \right)}}{\cos{\left(x - \pi \right)}}\right) \cos^{2}{\left(x \right)}}$$

                                      2                                     
                    /       2/x   pi\\                                      
                  2 |    sec |- - --||                                      
 /         1     \  |        \2   4 /|     2    /     1           sec(x)  \ 
-|1 + -----------| *|1 - ------------| *sec (x)*|----------- - -----------| 
 |       /    pi\|  |       2/x   pi\|          |   /    pi\      /    pi\| 
 |    sec|x - --||  |    sec |- + --||          |sec|x - --|   sec|x - --|| 
 \       \    2 //  \        \2   4 //          \   \    2 /      \    2 // 
----------------------------------------------------------------------------
                   /     1           sec(x)  \    2/    pi\                 
                 4*|----------- + -----------|*sec |x - --|                 
                   |   /    pi\      /    pi\|     \    2 /                 
                   |sec|x - --|   sec|x - --||                              
                   \   \    2 /      \    2 //

$$- \frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \left(- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \sec^{2}{\left(x \right)}}{4 \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$

     4    /         1     \  
 -sin (x)*|1 - -----------|  
          |       /    pi\|  
          |    sin|x + --||  
          \       \    2 //  
-----------------------------
/       /    pi\\    /    pi\
|1 + sin|x + --||*sin|x + --|
\       \    2 //    \    2 /

$$- \frac{\left(1 - \frac{1}{\sin{\left(x + \frac{\pi}{2} \right)}}\right) \sin^{4}{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \sin{\left(x + \frac{\pi}{2} \right)}}$$

    4    /      1   \ 
-sin (x)*|1 - ------| 
         \    cos(x)/ 
----------------------
 (1 + cos(x))*cos(x)

$$- \frac{\left(1 - \frac{1}{\cos{\left(x \right)}}\right) \sin^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}$$

                        /           /           /x\ \                    \    
                        |           |      2*cot|-| |                    |    
                        |           |           \2/ | /        2/x   pi\\|    
                        |           |1 + -----------|*|-1 + cot |- + --|||    
                        |           |           2/x\| \         \2   4 //|    
                      2 |           |    1 + cot |-||                    |    
   /          1      \  |    1      \            \2//                    |    
   |-1 + ------------| *|- ------ - -------------------------------------|    
   |        2/x   pi\|  \  cot(x)                     2                  /    
   |     cot |- + --||                                                        
   \         \2   4 //                                                        
------------------------------------------------------------------------------
                    /           /           /x\ \                    \        
                    |           |      2*cot|-| |                    |        
                    |           |           \2/ | /        2/x   pi\\|        
                    |           |1 + -----------|*|-1 + cot |- + --|||        
                    |           |           2/x\| \         \2   4 //|        
                  2 |           |    1 + cot |-||                    |        
/         1      \  |    1      \            \2//                    |    2   
|1 + ------------| *|- ------ + -------------------------------------|*cot (x)
|       2/x   pi\|  \  cot(x)                     2                  /        
|    cot |- + --||                                                            
\        \2   4 //

$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(- \frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} - \frac{1}{\cot{\left(x \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} - \frac{1}{\cot{\left(x \right)}}\right) \cot^{2}{\left(x \right)}}$$

                             /                       /          1      \\
                             |          (1 + sin(x))*|-1 + ------------||
                             |                       |        2/x   pi\||
                   2         |                       |     tan |- + --|||
/        2/x   pi\\     2    |                       \         \2   4 //|
|-1 + tan |- + --|| *tan (x)*|-tan(x) - --------------------------------|
\         \2   4 //          \                         2                /
-------------------------------------------------------------------------
                         /                       /          1      \\    
                         |          (1 + sin(x))*|-1 + ------------||    
                         |                       |        2/x   pi\||    
                       2 |                       |     tan |- + --|||    
     /       2/x   pi\\  |                       \         \2   4 //|    
     |1 + tan |- + --|| *|-tan(x) + --------------------------------|    
     \        \2   4 //  \                         2                /

$$\frac{\left(- \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$

                     2                                              
 /          2       \                        /       2            \ 
 |       cos (x)    |              2    4    |  2*sin (x)         | 
-|1 - --------------| *(1 + sin(x)) *sin (x)*|- --------- + sin(x)| 
 |         4/x   pi\|                        \   sin(2*x)         / 
 |    4*sin |- + --||                                               
 \          \2   4 //                                               
--------------------------------------------------------------------
                   /     2            \                             
                   |2*sin (x)         |    2                        
                   |--------- + sin(x)|*sin (2*x)                   
                   \ sin(2*x)         /

$$- \frac{\left(1 - \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right)^{2} \sin^{4}{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \sin{\left(x \right)}\right) \sin^{2}{\left(2 x \right)}}$$

                   2                                          
 /       2/x   pi\\              2    2                       
-|1 - cot |- + --|| *(1 + sin(x)) *tan (x)*(-tan(x) + sin(x)) 
 \        \2   4 //                                           
--------------------------------------------------------------
                     4*(sin(x) + tan(x))

$$- \frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{4 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}$$

                             /          /           /x\ \                    \
                             |          |      2*tan|-| |                    |
                             |          |           \2/ | /          1      \|
                             |          |1 + -----------|*|-1 + ------------||
                             |          |           2/x\| |        2/x   pi\||
                   2         |          |    1 + tan |-|| |     tan |- + --|||
/        2/x   pi\\     2    |          \            \2// \         \2   4 //|
|-1 + tan |- + --|| *tan (x)*|-tan(x) - -------------------------------------|
\         \2   4 //          \                            2                  /
------------------------------------------------------------------------------
                        /          /           /x\ \                    \     
                        |          |      2*tan|-| |                    |     
                        |          |           \2/ | /          1      \|     
                        |          |1 + -----------|*|-1 + ------------||     
                        |          |           2/x\| |        2/x   pi\||     
                      2 |          |    1 + tan |-|| |     tan |- + --|||     
    /       2/x   pi\\  |          \            \2// \         \2   4 //|     
    |1 + tan |- + --|| *|-tan(x) + -------------------------------------|     
    \        \2   4 //  \                            2                  /

$$\frac{\left(- \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$

    4/    pi\ /      1   \ 
-cos |x - --|*|1 - ------| 
     \    2 / \    cos(x)/ 
---------------------------
    (1 + cos(x))*cos(x)

$$- \frac{\left(1 - \frac{1}{\cos{\left(x \right)}}\right) \cos^{4}{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}$$

                                      2                          
                     /           /x\ \  /                  /x\ \ 
                   2 |      2*cot|-| |  |             2*cot|-| | 
 /       2/x   pi\\  |           \2/ |  |    1             \2/ | 
-|1 - cot |- + --|| *|1 + -----------| *|- ------ + -----------| 
 \        \2   4 //  |           2/x\|  |  cot(x)          2/x\| 
                     |    1 + cot |-||  |           1 + cot |-|| 
                     \            \2//  \                   \2// 
-----------------------------------------------------------------
                   /                /x\ \                        
                   |           2*cot|-| |                        
                   |  1             \2/ |    2                   
                 4*|------ + -----------|*cot (x)                
                   |cot(x)          2/x\|                        
                   |         1 + cot |-||                        
                   \                 \2//

$$- \frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(- \frac{1}{\cot{\left(x \right)}} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{4 \left(\frac{1}{\cot{\left(x \right)}} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \cot^{2}{\left(x \right)}}$$

       4/x\    6/x\ /      1   \
-16*cot |-|*sin |-|*|1 - ------|
        \2/     \2/ \    cos(x)/
--------------------------------
               /        2/x\\   
  (1 + cos(x))*|-1 + cot |-||   
               \         \2//

$$- \frac{16 \left(1 - \frac{1}{\cos{\left(x \right)}}\right) \sin^{6}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}$$

                            /                 2/x   pi\\
                  2         |          1 - tan |- + --||
/         1      \     2    |                  \2   4 /|
|1 - ------------| *tan (x)*|-tan(x) - ----------------|
|       2/x   pi\|          |                 2/x   pi\|
|    tan |- + --||          |          1 + tan |- + --||
\        \2   4 //          \                  \2   4 //
--------------------------------------------------------
                        /                 2/x   pi\\    
                      2 |          1 - tan |- + --||    
    /         1      \  |                  \2   4 /|    
    |1 + ------------| *|-tan(x) + ----------------|    
    |       2/x   pi\|  |                 2/x   pi\|    
    |    tan |- + --||  |          1 + tan |- + --||    
    \        \2   4 //  \                  \2   4 //

$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \tan{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \tan{\left(x \right)}\right)}$$

                        2                         2                                     
 /      ___    /    pi\\  /       ___    /    pi\\               2    2    /      1   \ 
-|1 + \/ 2 *sin|x + --|| *|-1 + \/ 2 *cos|x + --|| *(-1 + sin(x)) *sin (x)*|1 - ------| 
 \             \    4 //  \              \    4 //                         \    cos(x)/ 
----------------------------------------------------------------------------------------
                                                   5                                    
                                 4*(1 + cos(x))*cos (x)

$$- \frac{\left(1 - \frac{1}{\cos{\left(x \right)}}\right) \left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\sin{\left(x \right)} - 1\right)^{2} \sin^{2}{\left(x \right)}}{4 \left(\cos{\left(x \right)} + 1\right) \cos^{5}{\left(x \right)}}$$

                    /           2/x\\         
                    |    1 + tan |-||         
               4/x\ |            \2/|         
        -16*tan |-|*|1 - -----------|         
                \2/ |           2/x\|         
                    |    1 - tan |-||         
                    \            \2//         
----------------------------------------------
                             /           2/x\\
             3               |    1 - tan |-||
/       2/x\\  /       2/x\\ |            \2/|
|1 + tan |-|| *|1 - tan |-||*|1 + -----------|
\        \2//  \        \2// |           2/x\|
                             |    1 + tan |-||
                             \            \2//

$$- \frac{16 \left(1 - \frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan^{4}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$

             /            /pi    \\
             |         csc|-- - x||
   2/pi    \ |  1         \2     /|
csc |-- - x|*|------ - -----------|
    \2     / \csc(x)      csc(x)  /
-----------------------------------
  /              /pi    \\         
  |           csc|-- - x||         
  |    1         \2     /|    4    
  |- ------ - -----------|*csc (x) 
  \  csc(x)      csc(x)  /

$$\frac{\left(- \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\left(- \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \csc^{4}{\left(x \right)}}$$

   2       2      /                sin(2*x)\
sin (x)*sin (2*x)*|-sin(pi + x) - ---------|
                  |                    2   |
                  \               2*cos (x)/
--------------------------------------------
     /   sin(2*x)              \    4       
   4*|- --------- + sin(pi + x)|*cos (x)    
     |       2                 |            
     \  2*cos (x)              /

$$\frac{\left(- \frac{\sin{\left(2 x \right)}}{2 \cos^{2}{\left(x \right)}} - \sin{\left(x + \pi \right)}\right) \sin^{2}{\left(x \right)} \sin^{2}{\left(2 x \right)}}{4 \left(- \frac{\sin{\left(2 x \right)}}{2 \cos^{2}{\left(x \right)}} + \sin{\left(x + \pi \right)}\right) \cos^{4}{\left(x \right)}}$$

                        /                  2/x   pi\\    
                      2 |           1 - tan |- + --||    
    /       2/x   pi\\  |    1              \2   4 /|    
    |1 - cot |- + --|| *|- ------ - ----------------|    
    \        \2   4 //  |  cot(x)          2/x   pi\|    
                        |           1 + tan |- + --||    
                        \                   \2   4 //    
---------------------------------------------------------
                    /                  2/x   pi\\        
                  2 |           1 - tan |- + --||        
/       2/x   pi\\  |    1              \2   4 /|    2   
|1 + cot |- + --|| *|- ------ + ----------------|*cot (x)
\        \2   4 //  |  cot(x)          2/x   pi\|        
                    |           1 + tan |- + --||        
                    \                   \2   4 //

$$\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \frac{1}{\cot{\left(x \right)}}\right)}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \cot^{2}{\left(x \right)}}$$

                              /                                  /        2/x   pi\\\ 
                              |                                  |     sec |- - --||| 
                              |                /         1     \ |         \2   4 /|| 
                    2         |                |1 + -----------|*|-1 + ------------|| 
 /        2/x   pi\\          |                |       /    pi\| |        2/x   pi\|| 
 |     sec |- + --||          |                |    sec|x - --|| |     sec |- + --||| 
 |         \2   4 /|     2    |     sec(x)     \       \    2 // \         \2   4 //| 
 |-1 + ------------| *sec (x)*|- ----------- - -------------------------------------| 
 |        2/x   pi\|          |     /    pi\                     2                  | 
 |     sec |- - --||          |  sec|x - --|                                        | 
 \         \2   4 //          \     \    2 /                                        / 
--------------------------------------------------------------------------------------
                    /                  /        2/x   pi\\              \             
                    |                  |     sec |- - --||              |             
                    |/         1     \ |         \2   4 /|              |             
                  2 ||1 + -----------|*|-1 + ------------|              |             
/       2/x   pi\\  ||       /    pi\| |        2/x   pi\|              |             
|    sec |- + --||  ||    sec|x - --|| |     sec |- + --||              |             
|        \2   4 /|  |\       \    2 // \         \2   4 //      sec(x)  |    2/    pi\
|1 + ------------| *|------------------------------------- - -----------|*sec |x - --|
|       2/x   pi\|  |                  2                        /    pi\|     \    2 /
|    sec |- - --||  |                                        sec|x - --||             
\        \2   4 //  \                                           \    2 //

$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(- \frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - 1\right)}{2} - \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(x \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(\frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - 1\right)}{2} - \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$

           2    /  1         sec(x)  \       
        sec (x)*|------ - -----------|       
                |csc(x)      /    pi\|       
                |         sec|x - --||       
                \            \    2 //       
---------------------------------------------
/    1         sec(x)  \    2       2/    pi\
|- ------ - -----------|*csc (x)*sec |x - --|
|  csc(x)      /    pi\|             \    2 /
|           sec|x - --||                     
\              \    2 //

$$\frac{\left(- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec^{2}{\left(x \right)}}{\left(- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \csc^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$

                                  /                                  /        2/x   pi\\\
                                  |                                  |     cos |- + --|||
                                  |                /       /    pi\\ |         \2   4 /||
                   2              |                |1 + cos|x - --||*|-1 + ------------||
/        2/x   pi\\               |     /    pi\   \       \    2 // |        2/x   pi\||
|     cos |- - --||               |  cos|x - --|                     |     cos |- - --|||
|         \2   4 /|     2/    pi\ |     \    2 /                     \         \2   4 //|
|-1 + ------------| *cos |x - --|*|- ----------- - -------------------------------------|
|        2/x   pi\|      \    2 / \     cos(x)                       2                  /
|     cos |- + --||                                                                      
\         \2   4 //                                                                      
-----------------------------------------------------------------------------------------
                        /                  /        2/x   pi\\              \            
                        |                  |     cos |- + --||              |            
                        |/       /    pi\\ |         \2   4 /|              |            
                      2 ||1 + cos|x - --||*|-1 + ------------|              |            
    /       2/x   pi\\  |\       \    2 // |        2/x   pi\|      /    pi\|            
    |    cos |- - --||  |                  |     cos |- - --||   cos|x - --||            
    |        \2   4 /|  |                  \         \2   4 //      \    2 /|    2       
    |1 + ------------| *|------------------------------------- - -----------|*cos (x)    
    |       2/x   pi\|  \                  2                        cos(x)  /            
    |    cos |- + --||                                                                   
    \        \2   4 //

$$\frac{\left(- \frac{\left(-1 + \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}{2} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - 1\right)^{2} \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(\frac{\left(-1 + \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}{2} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \cos^{2}{\left(x \right)}}$$

  -(1 - sec(x))*sec(x)   
-------------------------
/      1   \    4/    pi\
|1 + ------|*sec |x - --|
\    sec(x)/     \    2 /

$$- \frac{\left(1 - \sec{\left(x \right)}\right) \sec{\left(x \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{4}{\left(x - \frac{\pi}{2} \right)}}$$

                   2                                                               
 /       2/x   pi\\                                  /     /    pi\              \ 
 |    cos |- + --||                   2              |  cos|x - --|              | 
 |        \2   4 /|  /       /    pi\\     2/    pi\ |     \    2 /      /    pi\| 
-|1 - ------------| *|1 + cos|x - --|| *cos |x - --|*|- ----------- + cos|x - --|| 
 |       2/x   pi\|  \       \    2 //      \    2 / \     cos(x)        \    2 // 
 |    cos |- - --||                                                                
 \        \2   4 //                                                                
-----------------------------------------------------------------------------------
                         /   /    pi\              \                               
                         |cos|x - --|              |                               
                         |   \    2 /      /    pi\|    2                          
                       4*|----------- + cos|x - --||*cos (x)                       
                         \   cos(x)        \    2 //

$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2} \left(\cos{\left(x - \frac{\pi}{2} \right)} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{4 \left(\cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}$$

                                   2                                     
               /        2/x   pi\ \               /            /pi    \\ 
             2 |     csc |- + --| |               |         csc|-- - x|| 
 /      1   \  |         \2   4 / |     2/pi    \ |  1         \2     /| 
-|1 + ------| *|1 - --------------| *csc |-- - x|*|------ - -----------| 
 \    csc(x)/  |       2/  x   pi\|      \2     / \csc(x)      csc(x)  / 
               |    csc |- - + --||                                      
               \        \  2   4 //                                      
-------------------------------------------------------------------------
                       /            /pi    \\                            
                       |         csc|-- - x||                            
                       |  1         \2     /|    2                       
                     4*|------ + -----------|*csc (x)                    
                       \csc(x)      csc(x)  /

$$- \frac{\left(1 - \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \left(- \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{4 \left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc^{2}{\left(x \right)}}$$

                     /           2/x\ \         
                     |    1 + cot |-| |         
                4/x\ |            \2/ |         
         -16*cot |-|*|1 - ------------|         
                 \2/ |            2/x\|         
                     |    -1 + cot |-||         
                     \             \2//         
------------------------------------------------
               /            2/x\\               
             3 |    -1 + cot |-||               
/       2/x\\  |             \2/| /        2/x\\
|1 + cot |-|| *|1 + ------------|*|-1 + cot |-||
\        \2//  |           2/x\ | \         \2//
               |    1 + cot |-| |               
               \            \2/ /

$$- \frac{16 \left(1 - \frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{\cot^{2}{\left(\frac{x}{2} \right)} - 1}\right) \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$

                             /                       /        2/x   pi\\\
                   2         |          (1 + sin(x))*|-1 + cot |- + --|||
/        2/x   pi\\     2    |                       \         \2   4 //|
|-1 + tan |- + --|| *tan (x)*|-tan(x) - --------------------------------|
\         \2   4 //          \                         2                /
-------------------------------------------------------------------------
                         /                       /        2/x   pi\\\    
                       2 |          (1 + sin(x))*|-1 + cot |- + --|||    
     /       2/x   pi\\  |                       \         \2   4 //|    
     |1 + tan |- + --|| *|-tan(x) + --------------------------------|    
     \        \2   4 //  \                         2                /

$$\frac{\left(- \frac{\left(\sin{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(\frac{\left(\sin{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$

                                 /                           /           2       \\
                                 |                           |        cos (x)    ||
                       2         |              (1 + sin(x))*|-1 + --------------||
  /          4/x   pi\\          |                           |          4/x   pi\||
  |     4*sin |- + --||          |       2                   |     4*sin |- + --|||
  |           \2   4 /|     4    |  2*sin (x)                \           \2   4 //|
4*|-1 + --------------| *sin (x)*|- --------- - ----------------------------------|
  |           2       |          \   sin(2*x)                   2                 /
  \        cos (x)    /                                                            
-----------------------------------------------------------------------------------
                        /             /           2       \            \           
                        |             |        cos (x)    |            |           
                      2 |(1 + sin(x))*|-1 + --------------|            |           
  /         4/x   pi\\  |             |          4/x   pi\|            |           
  |    4*sin |- + --||  |             |     4*sin |- + --||        2   |           
  |          \2   4 /|  |             \           \2   4 //   2*sin (x)|    2      
  |1 + --------------| *|---------------------------------- - ---------|*sin (2*x) 
  |          2       |  \                2                     sin(2*x)/           
  \       cos (x)    /

$$\frac{4 \left(- \frac{\left(-1 + \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} - \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(x \right)}} - 1\right)^{2} \sin^{4}{\left(x \right)}}{\left(\frac{\left(-1 + \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} - \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(2 x \right)}}$$

       6/x\    4/x\ /      1   \
-16*cos |-|*tan |-|*|1 - ------|
        \2/     \2/ \    cos(x)/
--------------------------------
   /       2/x\\                
   |1 - tan |-||*(1 + cos(x))   
   \        \2//

$$- \frac{16 \left(1 - \frac{1}{\cos{\left(x \right)}}\right) \cos^{6}{\left(\frac{x}{2} \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$

                             /                  2/x   pi\\
                   2         |          -1 + cot |- + --||
/        2/x   pi\\     2    |                   \2   4 /|
|-1 + tan |- + --|| *tan (x)*|-tan(x) - -----------------|
\         \2   4 //          |                  2/x   pi\|
                             |           1 + cot |- + --||
                             \                   \2   4 //
----------------------------------------------------------
                        /                  2/x   pi\\     
                      2 |          -1 + cot |- + --||     
    /       2/x   pi\\  |                   \2   4 /|     
    |1 + tan |- + --|| *|-tan(x) + -----------------|     
    \        \2   4 //  |                  2/x   pi\|     
                        |           1 + cot |- + --||     
                        \                   \2   4 //

$$\frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \tan{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$

                        /                   2/x   pi\\    
                      2 |           -1 + cot |- + --||    
   /          1      \  |    1               \2   4 /|    
   |-1 + ------------| *|- ------ - -----------------|    
   |        2/x   pi\|  |  cot(x)           2/x   pi\|    
   |     cot |- + --||  |            1 + cot |- + --||    
   \         \2   4 //  \                    \2   4 //    
----------------------------------------------------------
                    /                   2/x   pi\\        
                  2 |           -1 + cot |- + --||        
/         1      \  |    1               \2   4 /|    2   
|1 + ------------| *|- ------ + -----------------|*cot (x)
|       2/x   pi\|  |  cot(x)           2/x   pi\|        
|    cot |- + --||  |            1 + cot |- + --||        
\        \2   4 //  \                    \2   4 //

$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(- \frac{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \frac{1}{\cot{\left(x \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\frac{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} - \frac{1}{\cot{\left(x \right)}}\right) \cot^{2}{\left(x \right)}}$$

    2    /  1      sec(x)\ 
 sec (x)*|------ - ------| 
         \csc(x)   csc(x)/ 
---------------------------
/    1      sec(x)\    4   
|- ------ - ------|*csc (x)
\  csc(x)   csc(x)/

$$\frac{\left(- \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec^{2}{\left(x \right)}}{\left(- \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \csc^{4}{\left(x \right)}}$$

 /       /pi    \\    /pi    \ 
-|1 - csc|-- - x||*csc|-- - x| 
 \       \2     //    \2     / 
-------------------------------
   /         1     \    4      
   |1 + -----------|*csc (x)   
   |       /pi    \|           
   |    csc|-- - x||           
   \       \2     //

$$- \frac{\left(1 - \csc{\left(- x + \frac{\pi}{2} \right)}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{4}{\left(x \right)}}$$

   2       2      /     sin(2*x)            \
sin (x)*sin (2*x)*|- -------------- + sin(x)|
                  |       2/    pi\         |
                  |  2*sin |x + --|         |
                  \        \    2 /         /
---------------------------------------------
    /             sin(2*x)   \    4/    pi\  
  4*|-sin(x) - --------------|*sin |x + --|  
    |               2/    pi\|     \    2 /  
    |          2*sin |x + --||               
    \                \    2 //

$$\frac{\left(\sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}\right) \sin^{2}{\left(x \right)} \sin^{2}{\left(2 x \right)}}{4 \left(- \sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}\right) \sin^{4}{\left(x + \frac{\pi}{2} \right)}}$$

    4                        
-sin (x)*(1 - sec(x))*sec(x) 
-----------------------------
          1 + cos(x)

$$- \frac{\left(1 - \sec{\left(x \right)}\right) \sin^{4}{\left(x \right)} \sec{\left(x \right)}}{\cos{\left(x \right)} + 1}$$

             /     /    pi\              \
             |  cos|x - --|              |
   4/    pi\ |     \    2 /      /    pi\|
cos |x - --|*|- ----------- + cos|x - --||
    \    2 / \     cos(x)        \    2 //
------------------------------------------
  /                   /    pi\\           
  |                cos|x - --||           
  |     /    pi\      \    2 /|    2      
  |- cos|x - --| - -----------|*cos (x)   
  \     \    2 /      cos(x)  /

$$\frac{\left(\cos{\left(x - \frac{\pi}{2} \right)} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos^{4}{\left(x - \frac{\pi}{2} \right)}}{\left(- \cos{\left(x - \frac{\pi}{2} \right)} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}$$

                                    /                             /         2/x   pi\ \\
                                    |                             |      csc |- + --| ||
                                    |                /      1   \ |          \2   4 / ||
                     2              |                |1 + ------|*|-1 + --------------||
/        2/  x   pi\\               |     /pi    \   \    csc(x)/ |        2/  x   pi\||
|     csc |- - + --||               |  csc|-- - x|                |     csc |- - + --|||
|         \  2   4 /|     2/pi    \ |     \2     /                \         \  2   4 //|
|-1 + --------------| *csc |-- - x|*|- ----------- - ----------------------------------|
|         2/x   pi\ |      \2     / \     csc(x)                     2                 /
|      csc |- + --| |                                                                   
\          \2   4 / /                                                                   
----------------------------------------------------------------------------------------
                          /             /         2/x   pi\ \              \            
                          |             |      csc |- + --| |              |            
                          |/      1   \ |          \2   4 / |              |            
                        2 ||1 + ------|*|-1 + --------------|              |            
    /       2/  x   pi\\  |\    csc(x)/ |        2/  x   pi\|      /pi    \|            
    |    csc |- - + --||  |             |     csc |- - + --||   csc|-- - x||            
    |        \  2   4 /|  |             \         \  2   4 //      \2     /|    2       
    |1 + --------------| *|---------------------------------- - -----------|*csc (x)    
    |        2/x   pi\ |  \                2                       csc(x)  /            
    |     csc |- + --| |                                                                
    \         \2   4 / /

$$\frac{\left(- \frac{\left(-1 + \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right)}{2} - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \left(\frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - 1\right)^{2} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\left(\frac{\left(-1 + \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right)}{2} - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \left(\frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \csc^{2}{\left(x \right)}}$$

                   2                                          
 /         1      \              2    2                       
-|1 - ------------| *(1 + sin(x)) *tan (x)*(-tan(x) + sin(x)) 
 |       2/x   pi\|                                           
 |    tan |- + --||                                           
 \        \2   4 //                                           
--------------------------------------------------------------
                     4*(sin(x) + tan(x))

$$- \frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{4 \left(\sin{\left(x \right)} + \tan{\left(x \right)}\right)}$$

           2    /     1        sec(x)\       
        sec (x)*|----------- - ------|       
                |   /pi    \   csc(x)|       
                |sec|-- - x|         |       
                \   \2     /         /       
---------------------------------------------
/       1        sec(x)\    2       2/pi    \
|- ----------- - ------|*csc (x)*sec |-- - x|
|     /pi    \   csc(x)|             \2     /
|  sec|-- - x|         |                     
\     \2     /         /

$$\frac{\left(\frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \sec^{2}{\left(x \right)}}{\left(- \frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \csc^{2}{\left(x \right)} \sec^{2}{\left(- x + \frac{\pi}{2} \right)}}$$

sec(x)^2*(1/sec(pi/2 - x) - sec(x)/csc(x))/((-1/sec(pi/2 - x) - sec(x)/csc(x))*csc(x)^2*sec(pi/2 - x)^2)

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Least common denominator cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))*(cot(x-pi/2)-cos(x+pi/2))

The solution

Least common denominator cos(x-pi/2)^(2)cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2))(cot(x-pi/2)-cos(x+pi/2))