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How to use it?
Least common denominator:
(x-2*cos(x)/sin(x)+2*x*cos(x)^2/sin(x)^2)/sin(x)
(x^2-8*x/9+493/1000)*(x-x^2/2)/3
((x+2)/(2-x)-(2-x)/(x+2))*((x-2)^2)
w2/212*a+w-144*a2/212*a+w
Factor polynomial:
y^3/3+2*x*y^2
y^2+x*y+x^2+2*x+2*y
y^2+625
y^2+5^3
Factor squared:
-y^4+2*y^2-9
-y^4+2*y^2-7
y^4-3*y^2+5
y^4+2*y^2+4
How do you in partial fractions?:
(x-6)/(x^2-36)
1/(x^2-4x+4)
-1/(x+1)^2
-10*x/(x^2-16)^2
Identical expressions
(x- two *cos(x)/sin(x)+ two *x*cos(x)^ two /sin(x)^ two)/sin(x)
(x minus 2 multiply by co sinus of e of (x) divide by sinus of (x) plus 2 multiply by x multiply by co sinus of e of (x) squared divide by sinus of (x) squared ) divide by sinus of (x)
(x minus two multiply by co sinus of e of (x) divide by sinus of (x) plus two multiply by x multiply by co sinus of e of (x) to the power of two divide by sinus of (x) to the power of two) divide by sinus of (x)
(x-2*cos(x)/sin(x)+2*x*cos(x)2/sin(x)2)/sin(x)
x-2*cosx/sinx+2*x*cosx2/sinx2/sinx
(x-2*cos(x)/sin(x)+2*x*cos(x)²/sin(x)²)/sin(x)
(x-2*cos(x)/sin(x)+2*x*cos(x) to the power of 2/sin(x) to the power of 2)/sin(x)
(x-2cos(x)/sin(x)+2xcos(x)^2/sin(x)^2)/sin(x)
(x-2cos(x)/sin(x)+2xcos(x)2/sin(x)2)/sin(x)
x-2cosx/sinx+2xcosx2/sinx2/sinx
x-2cosx/sinx+2xcosx^2/sinx^2/sinx
(x-2*cos(x) divide by sin(x)+2*x*cos(x)^2 divide by sin(x)^2) divide by sin(x)
Similar expressions
(x-2*cos(x)/sin(x)-2*x*cos(x)^2/sin(x)^2)/sin(x)
(x+2*cos(x)/sin(x)+2*x*cos(x)^2/sin(x)^2)/sin(x)
(x-2*cosx/sinx+2*x*cosx^2/sinx^2)/sinx

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

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

The solution

You have entered [src]

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

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

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

General simplification [src]

            3*x   x*cos(2*x)
-sin(2*x) + --- + ----------
             2        2     
----------------------------
             3              
          sin (x)

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

(-sin(2*x) + 3*x/2 + x*cos(2*x)/2)/sin(x)^3

Rational denominator [src]

   2                                    2          
sin (x)*(-2*cos(x) + x*sin(x)) + 2*x*cos (x)*sin(x)
---------------------------------------------------
                         4                         
                      sin (x)

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

(sin(x)^2*(-2*cos(x) + x*sin(x)) + 2*x*cos(x)^2*sin(x))/sin(x)^4

Numerical answer [src]

(x - 2.0*cos(x)/sin(x) + 2.0*x*cos(x)^2/sin(x)^2)/sin(x)

(x - 2.0*cos(x)/sin(x) + 2.0*x*cos(x)^2/sin(x)^2)/sin(x)

Powers [src]

    /                      2                     \
    |        / I*x    -I*x\                      |
    |        |e      e    |                      |
    |    8*x*|---- + -----|        / I*x    -I*x\|
    |        \ 2       2  /    2*I*\e    + e    /|
2*I*|x - ------------------- - ------------------|
    |                     2         -I*x    I*x  |
    |     /   -I*x    I*x\       - e     + e     |
    \     \- e     + e   /                       /
--------------------------------------------------
                     -I*x    I*x                  
                  - e     + e

$$\frac{2 i \left(- \frac{8 x \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{\left(e^{i x} - e^{- i x}\right)^{2}} + x - \frac{2 i \left(e^{i x} + e^{- i x}\right)}{e^{i x} - e^{- i x}}\right)}{e^{i x} - e^{- i x}}$$

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

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

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

Combinatorics [src]

     2                               2   
x*sin (x) - 2*cos(x)*sin(x) + 2*x*cos (x)
-----------------------------------------
                    3                    
                 sin (x)

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

(x*sin(x)^2 - 2*cos(x)*sin(x) + 2*x*cos(x)^2)/sin(x)^3

Combining rational expressions [src]

                                       2   
(-2*cos(x) + x*sin(x))*sin(x) + 2*x*cos (x)
-------------------------------------------
                     3                     
                  sin (x)

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

((-2*cos(x) + x*sin(x))*sin(x) + 2*x*cos(x)^2)/sin(x)^3

Common denominator [src]

     2                               2   
x*sin (x) - 2*cos(x)*sin(x) + 2*x*cos (x)
-----------------------------------------
                    3                    
                 sin (x)

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

(x*sin(x)^2 - 2*cos(x)*sin(x) + 2*x*cos(x)^2)/sin(x)^3

Assemble expression [src]

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

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

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

Expand expression [src]

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

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

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

Trigonometric part [src]

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

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

/      /x\        /x   pi\        2/x   pi\          2/x\   \                   
|   sec|-|     sec|- - --|   x*sec |- - --|     x*sec |-|   |                   
|      \2/        \2   2 /         \2   2 /           \2/   |    /x\    /x   pi\
|----------- - ----------- + -------------- + --------------|*sec|-|*sec|- - --|
|   /x   pi\         /x\            2/x\           2/x   pi\|    \2/    \2   2 /
|sec|- - --|      sec|-|       2*sec |-|      2*sec |- - --||                   
\   \2   2 /         \2/             \2/            \2   2 //                   
--------------------------------------------------------------------------------
                                       2

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

              /                                   2\
              |            2/x\     /        2/x\\ |
              |    -1 + cot |-|   x*|-1 + cot |-|| |
/       2/x\\ |             \2/     \         \2// |
|1 + cot |-||*|x - ------------ + -----------------|
\        \2// |          /x\               2/x\    |
              |       cot|-|          2*cot |-|    |
              \          \2/                \2/    /
----------------------------------------------------
                           /x\                      
                      2*cot|-|                      
                           \2/

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

              /                                 2\
              |           2/x\     /       2/x\\ |
              |    1 - tan |-|   x*|1 - tan |-|| |
/       2/x\\ |            \2/     \        \2// |
|1 + tan |-||*|x - ----------- + ----------------|
\        \2// |          /x\             2/x\    |
              |       tan|-|        2*tan |-|    |
              \          \2/              \2/    /
--------------------------------------------------
                          /x\                     
                     2*tan|-|                     
                          \2/

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

                    2     
    sin(2*x)   x*sin (2*x)
x - -------- + -----------
       2             4    
    sin (x)     2*sin (x) 
--------------------------
          sin(x)

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

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

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

/                    2/x\          4/x\            \       
|               2*sin |-|   2*x*sin |-|        2   |       
|    sin(x)           \2/           \2/   x*sin (x)|       
|- ---------- + --------- + ----------- + ---------|*sin(x)
|  1 - cos(x)     sin(x)         2             4/x\|       
|                             sin (x)     8*sin |-||       
\                                               \2//       
-----------------------------------------------------------
                 (1 - cos(x))*(1 + cos(x))

$$\frac{\left(\frac{2 x \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \sin^{2}{\left(x \right)}}{8 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$

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

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

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

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

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

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

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

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

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

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

/      2        2*x  \       
|x - ------ + -------|*csc(x)
|    tan(x)      2   |       
\             tan (x)/

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

/                            2/x\         \       
|                       x*tan |-|         |       
|    1          x             \2/      /x\|    /x\
|- ------ + --------- + --------- + tan|-||*tan|-|
|     /x\        2/x\       2          \2/|    \2/
|  tan|-|   2*tan |-|                     |       
\     \2/         \2/                     /       
--------------------------------------------------
                         2/x\                     
                    2*sin |-|                     
                          \2/

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

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

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

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

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

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

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

                            2/x\         
                       x*tan |-|         
    1          x             \2/      /x\
- ------ + --------- + --------- + tan|-|
     /x\        2/x\       2          \2/
  tan|-|   2*tan |-|                     
     \2/         \2/                     
-----------------------------------------
                  2/x\    /x\            
             2*cos |-|*tan|-|            
                   \2/    \2/

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

     2/x\                      4/x\            
2*sin |-|               2*x*sin |-|        2   
      \2/     sin(x)            \2/   x*sin (x)
--------- - --------- + ----------- + ---------
  sin(x)         2/x\        2             4/x\
            2*sin |-|     sin (x)     8*sin |-|
                  \2/                       \2/
-----------------------------------------------
                     sin(x)

$$\frac{\frac{2 x \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \sin^{2}{\left(x \right)}}{8 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(\frac{x}{2} \right)}}}{\sin{\left(x \right)}}$$

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

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

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

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

/   /pi   x\         /x\          2/pi   x\          2/x\   \                   
|csc|-- - -|      csc|-|     x*csc |-- - -|     x*csc |-|   |                   
|   \2    2/         \2/           \2    2/           \2/   |    /x\    /pi   x\
|----------- - ----------- + -------------- + --------------|*csc|-|*csc|-- - -|
|      /x\        /pi   x\          2/x\           2/pi   x\|    \2/    \2    2/
|   csc|-|     csc|-- - -|     2*csc |-|      2*csc |-- - -||                   
\      \2/        \2    2/           \2/            \2    2//                   
--------------------------------------------------------------------------------
                                       2

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

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

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

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

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

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

                    2/x\          4/x\            
               2*sin |-|   2*x*sin |-|        2   
    sin(x)           \2/           \2/   x*sin (x)
- ---------- + --------- + ----------- + ---------
  1 - cos(x)     sin(x)         2             4/x\
                             sin (x)     8*sin |-|
                                               \2/
--------------------------------------------------
                      sin(x)

$$\frac{\frac{2 x \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \sin^{2}{\left(x \right)}}{8 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}}{\sin{\left(x \right)}}$$

/                      2   \       
|    2*csc(x)   2*x*csc (x)|       
|x - -------- + -----------|*csc(x)
|     sec(x)         2     |       
\                 sec (x)  /

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

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

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

               /                                   2/x\\
             2 |                              x*cot |-||
/       2/x\\  |  1         /x\       x             \2/|
|1 + cot |-|| *|------ - cot|-| + --------- + ---------|
\        \4//  |   /x\      \2/        2/x\       2    |
               |cot|-|            2*cot |-|            |
               \   \2/                  \2/            /
--------------------------------------------------------
                         /x\    2/x\                    
                    8*cot|-|*cot |-|                    
                         \2/     \4/

$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\frac{x \cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{x}{2 \cot^{2}{\left(\frac{x}{2} \right)}} - \cot{\left(\frac{x}{2} \right)} + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)}{8 \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}$$

               /                            2/x\         \       
             2 |                       x*tan |-|         |       
/       2/x\\  |    1          x             \2/      /x\|    /x\
|1 + tan |-|| *|- ------ + --------- + --------- + tan|-||*tan|-|
\        \4//  |     /x\        2/x\       2          \2/|    \2/
               |  tan|-|   2*tan |-|                     |       
               \     \2/         \2/                     /       
-----------------------------------------------------------------
                                 2/x\                            
                            8*tan |-|                            
                                  \4/

$$\frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{x}{2 \tan^{2}{\left(\frac{x}{2} \right)}} + \tan{\left(\frac{x}{2} \right)} - \frac{1}{\tan{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}}{8 \tan^{2}{\left(\frac{x}{4} \right)}}$$

/     2/x\                      4/x\            \       
|2*sin |-|               2*x*sin |-|        2   |       
|      \2/     sin(x)            \2/   x*sin (x)|       
|--------- - --------- + ----------- + ---------|*sin(x)
|  sin(x)         2/x\        2             4/x\|       
|            2*sin |-|     sin (x)     8*sin |-||       
\                  \2/                       \2//       
--------------------------------------------------------
                      2/x\    2/pi   x\                 
                 4*sin |-|*sin |-- + -|                 
                       \2/     \2    2/

$$\frac{\left(\frac{2 x \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{x \sin^{2}{\left(x \right)}}{8 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(\frac{x}{2} \right)}}\right) \sin{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}$$

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

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

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

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

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

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

The solution

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