Mister Exam

# Least common denominator cos(a)/2*sin(2)-1-sin(a)/2*cos(2)-1

An expression to simplify:

### The solution

You have entered [src]
cos(a)              sin(a)
------*sin(2) - 1 - ------*cos(2) - 1
2                   2              
$$\left(- \frac{\sin{\left(a \right)}}{2} \cos{\left(2 \right)} + \left(\frac{\cos{\left(a \right)}}{2} \sin{\left(2 \right)} - 1\right)\right) - 1$$
(cos(a)/2)*sin(2) - 1 - sin(a)/2*cos(2) - 1
General simplification [src]
     sin(-2 + a)
-2 - -----------
2     
$$- \frac{\sin{\left(a - 2 \right)}}{2} - 2$$
-2 - sin(-2 + a)/2
Combining rational expressions [src]
-4 + cos(a)*sin(2) - cos(2)*sin(a)
----------------------------------
2                 
$$\frac{- \sin{\left(a \right)} \cos{\left(2 \right)} + \sin{\left(2 \right)} \cos{\left(a \right)} - 4}{2}$$
(-4 + cos(a)*sin(2) - cos(2)*sin(a))/2
Rational denominator [src]
-4 + cos(a)*sin(2) - cos(2)*sin(a)
----------------------------------
2                 
$$\frac{- \sin{\left(a \right)} \cos{\left(2 \right)} + \sin{\left(2 \right)} \cos{\left(a \right)} - 4}{2}$$
(-4 + cos(a)*sin(2) - cos(2)*sin(a))/2
Common denominator [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
-2.0 + 0.454648713412841*cos(a) + 0.208073418273571*sin(a)
-2.0 + 0.454648713412841*cos(a) + 0.208073418273571*sin(a)
Combinatorics [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
Powers [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
                        / I*a    -I*a\     / -2*I    2*I\
/   -2*I    2*I\ |e      e    |     |e       e   | /   -I*a    I*a\
I*\- e     + e   /*|---- + -----|   I*|----- + ----|*\- e     + e   /
\ 4       4  /     \  2      2  /
-2 - --------------------------------- + ---------------------------------
2                                   4                
$$- \frac{i \left(e^{2 i} - e^{- 2 i}\right) \left(\frac{e^{i a}}{4} + \frac{e^{- i a}}{4}\right)}{2} + \frac{i \left(\frac{e^{- 2 i}}{2} + \frac{e^{2 i}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{4} - 2$$
-2 - i*(-exp(-2*i) + exp(2*i))*(exp(i*a)/4 + exp(-i*a)/4)/2 + i*(exp(-2*i)/2 + exp(2*i)/2)*(-exp(-i*a) + exp(i*a))/4
Assemble expression [src]
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
-2 + cos(a)*sin(2)/2 - cos(2)*sin(a)/2
Expand expression [src]
     sin(a)      2
-2 + ------ - cos (1)*sin(a) + cos(1)*cos(a)*sin(1)
2                                           
$$- \sin{\left(a \right)} \cos^{2}{\left(1 \right)} + \frac{\sin{\left(a \right)}}{2} + \sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(a \right)} - 2$$
-2 + sin(a)/2 - cos(1)^2*sin(a) + cos(1)*cos(a)*sin(1)
Trigonometric part [src]
          /     a\
tan|-1 + -|
\     2/
-2 - ----------------
2/     a\
1 + tan |-1 + -|
\     2/
$$-2 - \frac{\tan{\left(\frac{a}{2} - 1 \right)}}{\tan^{2}{\left(\frac{a}{2} - 1 \right)} + 1}$$
             1
-2 - ------------------
/         pi\
2*sec|-2 + a - --|
\         2 /
$$-2 - \frac{1}{2 \sec{\left(a - 2 - \frac{\pi}{2} \right)}}$$
          /     a\
cot|-1 + -|
\     2/
-2 - ----------------
2/     a\
1 + cot |-1 + -|
\     2/
$$-2 - \frac{\cot{\left(\frac{a}{2} - 1 \right)}}{\cot^{2}{\left(\frac{a}{2} - 1 \right)} + 1}$$
        /         pi\
cos|-2 + a - --|
\         2 /
-2 - ----------------
2        
$$- \frac{\cos{\left(a - 2 - \frac{\pi}{2} \right)}}{2} - 2$$
     -sin(-2 + a) + sin(2 + a)      2/a\ /     1           2   \    /a\
-2 + ------------------------- - sin |-|*|----------- - sin (1)|*cot|-|
4                   \2/ |       2             |    \2/
\1 + tan (1)          /       
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} \right)} - 2$$
     (-1 - cos(4) + 2*cos(2))*sin(a)     2*cos(a)*sin(2)
-2 - ------------------------------- + -------------------
2                   2
1 - cos(4) + 2*(1 - cos(2))          2      sin (2)
4*sin (1) + -------
2
sin (1)
$$- \frac{\left(-1 + 2 \cos{\left(2 \right)} - \cos{\left(4 \right)}\right) \sin{\left(a \right)}}{- \cos{\left(4 \right)} + 1 + 2 \left(1 - \cos{\left(2 \right)}\right)^{2}} + \frac{2 \sin{\left(2 \right)} \cos{\left(a \right)}}{\frac{\sin^{2}{\left(2 \right)}}{\sin^{2}{\left(1 \right)}} + 4 \sin^{2}{\left(1 \right)}} - 2$$
                   2
/        2/a\\           2/a\
|-1 + cot |-||      4*cot |-|
\         \4//            \4/      /     1             1     \    /a\
--------------- - --------------   |----------- - -----------|*cot|-|
2                2   |       1             2   |    \2/
/       2/a\\    /       2/a\\    |1 + -------   1 + cot (1)|
|1 + cot |-||    |1 + cot |-||    |       2                 |
\        \4//    \        \4//    \    cot (1)              /
-2 + -------------------------------- - ----------------------------------
/       1   \                                  2/a\
|1 + -------|*cot(1)                    1 + cot |-|
|       2   |                                   \2/
\    cot (1)/                                                  
$$\frac{\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{4 \cot^{2}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}}{\left(1 + \frac{1}{\cot^{2}{\left(1 \right)}}\right) \cot{\left(1 \right)}} - 2 - \frac{\left(- \frac{1}{\cot^{2}{\left(1 \right)} + 1} + \frac{1}{1 + \frac{1}{\cot^{2}{\left(1 \right)}}}\right) \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}$$
     /             2                 \
|/       2/a\\           2/a\   |
||1 - tan |-||      4*tan |-|   |
|\        \4//            \4/   |          /     1             1     \    /a\
|-------------- - --------------|*tan(1)   |----------- - -----------|*tan|-|
|             2                2|          |       2             1   |    \2/
|/       2/a\\    /       2/a\\ |          |1 + tan (1)   1 + -------|
||1 + tan |-||    |1 + tan |-|| |          |                     2   |
\\        \4//    \        \4// /          \                  tan (1)/
-2 + ---------------------------------------- - ----------------------------------
2                                       2/a\
1 + tan (1)                             1 + tan |-|
\2/            
$$\frac{\left(\frac{\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} - 2 - \frac{\left(- \frac{1}{\frac{1}{\tan^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
            1                 1
-2 + --------------- - ---------------
2*csc(2)*sec(a)   2*csc(a)*sec(2)
$$-2 + \frac{1}{2 \csc{\left(2 \right)} \sec{\left(a \right)}} - \frac{1}{2 \csc{\left(a \right)} \sec{\left(2 \right)}}$$
     /     1             1     \
|----------- - -----------|*sin(a)
|       2             1   |
|1 + tan (1)   1 + -------|          /   2/a\      2/a\\
|                     2   |          |cos |-| - sin |-||*tan(1)
\                  tan (1)/          \    \2/       \2//
-2 - ---------------------------------- + --------------------------
2                                  2
1 + tan (1)        
$$\frac{\left(- \sin^{2}{\left(\frac{a}{2} \right)} + \cos^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(1 \right)}}{1 + \tan^{2}{\left(1 \right)}} - \frac{\left(- \frac{1}{\frac{1}{\tan^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin{\left(a \right)}}{2} - 2$$
            1                  1
---------------- - ----------------
2                2/    pi\
sec (1)          sec |1 - --|   /   1           1      \
1 + ------------           \    2 /   |------- - ------------|*sec(1)
2/    pi\   1 + ------------   |   2/a\      2/a   pi\|
sec |1 - --|            2         |sec |-|   sec |- - --||
\    2 /         sec (1)      \    \2/       \2   2 //
-2 - ----------------------------------- + -------------------------------
/    pi\               /         2      \
2*sec|a - --|               |      sec (1)   |    /    pi\
\    2 /               |1 + ------------|*sec|1 - --|
|       2/    pi\|    \    2 /
|    sec |1 - --||
\        \    2 //            
$$\frac{\left(- \frac{1}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(1 \right)}}{\left(1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}\right) \sec{\left(1 - \frac{\pi}{2} \right)}} - 2 - \frac{- \frac{1}{\frac{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}{\sec^{2}{\left(1 \right)}} + 1} + \frac{1}{1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}}}{2 \sec{\left(a - \frac{\pi}{2} \right)}}$$
         /       2/a\\                 /       2   \    /a\
|1 - tan |-||*tan(1)          \1 - tan (1)/*tan|-|
\        \2//                                  \2/
-2 + --------------------------- - ---------------------------
/       2   \ /       2/a\\   /       2   \ /       2/a\\
\1 + tan (1)/*|1 + tan |-||   \1 + tan (1)/*|1 + tan |-||
\        \2//                 \        \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(1 \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} - 2 - \frac{\left(1 - \tan^{2}{\left(1 \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(1 + \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
     sin(-2 + a)
-2 - -----------
2     
$$- \frac{\sin{\left(a - 2 \right)}}{2} - 2$$
     /      1               1      \
|------------- - -------------|*sin(a)
|         4              2    |
|    4*sin (1)        sin (2) |
|1 + ---------   1 + ---------|               2    /   2/pi   a\      2/a\\
|        2                4   |          2*sin (1)*|sin |-- + -| - sin |-||
\     sin (2)        4*sin (1)/                    \    \2    2/       \2//
-2 - -------------------------------------- + ----------------------------------
2                            /         4   \
|    4*sin (1)|
|1 + ---------|*sin(2)
|        2    |
\     sin (2) /             
$$\frac{2 \left(- \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}\right) \sin^{2}{\left(1 \right)}}{\left(1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}\right) \sin{\left(2 \right)}} - \frac{\left(- \frac{1}{\frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1} + \frac{1}{1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}}\right) \sin{\left(a \right)}}{2} - 2$$
        /        2/a\\                /        2   \    /a\
|-1 + cot |-||*cot(1)         \-1 + cot (1)/*cot|-|
\         \2//                                  \2/
-2 + --------------------------- - ---------------------------
/       2   \ /       2/a\\   /       2   \ /       2/a\\
\1 + cot (1)/*|1 + cot |-||   \1 + cot (1)/*|1 + cot |-||
\        \2//                 \        \2//
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(1 \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} - 2 - \frac{\left(-1 + \cot^{2}{\left(1 \right)}\right) \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(1 \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
     /       1                  1        \    /    pi\
|---------------- - ----------------|*cos|a - --|
|       2/    pi\            2      |    \    2 /
|    cos |1 - --|         cos (1)   |
|        \    2 /   1 + ------------|
|1 + ------------          2/    pi\|               /   2/a\      2/a   pi\\    /    pi\
|         2             cos |1 - --||               |cos |-| - cos |- - --||*cos|1 - --|
\      cos (1)              \    2 //               \    \2/       \2   2 //    \    2 /
-2 - ------------------------------------------------- + ------------------------------------
2                                /       2/    pi\\
|    cos |1 - --||
|        \    2 /|
|1 + ------------|*cos(1)
|         2      |
\      cos (1)   /             
$$\frac{\left(\cos^{2}{\left(\frac{a}{2} \right)} - \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}\right) \cos{\left(1 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}\right) \cos{\left(1 \right)}} - \frac{\left(- \frac{1}{\frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}} + 1} + \frac{1}{1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}}\right) \cos{\left(a - \frac{\pi}{2} \right)}}{2} - 2$$
               /    pi\             /    pi\
cos(a)*cos|2 - --|   cos(2)*cos|a - --|
\    2 /             \    2 /
-2 + ------------------ - ------------------
2                    2         
$$\frac{\cos{\left(a \right)} \cos{\left(2 - \frac{\pi}{2} \right)}}{2} - \frac{\cos{\left(2 \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{2} - 2$$
             1                   1
----------------- - -----------------
2/     pi\             2
csc |-1 + --|          csc (1)      /     1            1   \    /     pi\
\     2 /   1 + -------------   |------------ - -------|*csc|-1 + --|
1 + -------------          2/     pi\   |   2/pi   a\      2/a\|    \     2 /
2             csc |-1 + --|   |csc |-- - -|   csc |-||
csc (1)              \     2 /   \    \2    2/       \2//
-2 - ------------------------------------- + -------------------------------------
2*csc(a)                       /       2/     pi\\
|    csc |-1 + --||
|        \     2 /|
|1 + -------------|*csc(1)
|          2      |
\       csc (1)   /            
$$\frac{\left(\frac{1}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(-1 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}\right) \csc{\left(1 \right)}} - 2 - \frac{- \frac{1}{\frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1} + \frac{1}{1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}}}{2 \csc{\left(a \right)}}$$
              1                      1
-2 + -------------------- - --------------------
/    pi\               /    pi\
2*sec(a)*sec|2 - --|   2*sec(2)*sec|a - --|
\    2 /               \    2 /
$$-2 - \frac{1}{2 \sec{\left(2 \right)} \sec{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(a \right)} \sec{\left(2 - \frac{\pi}{2} \right)}}$$
              1                       1
-2 + -------------------- - ---------------------
/pi    \               /     pi\
2*csc(2)*csc|-- - a|   2*csc(a)*csc|-2 + --|
\2     /               \     2 /
$$-2 + \frac{1}{2 \csc{\left(2 \right)} \csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(a \right)} \csc{\left(-2 + \frac{\pi}{2} \right)}}$$
                                 /     1           2   \
|----------- - sin (1)|*sin(a)
|       2             |
-sin(-2 + a) + sin(2 + a)   \1 + tan (1)          /
-2 + ------------------------- - ------------------------------
4                             2               
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \frac{\left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \sin{\left(a \right)}}{2} - 2$$
     -sin(-2 + a) + sin(2 + a)      2/a\ /     1           2   \    /a\
-2 + ------------------------- - cos |-|*|----------- - sin (1)|*tan|-|
4                   \2/ |       2             |    \2/
\1 + tan (1)          /       
$$\frac{- \sin{\left(a - 2 \right)} + \sin{\left(a + 2 \right)}}{4} - \left(- \sin^{2}{\left(1 \right)} + \frac{1}{1 + \tan^{2}{\left(1 \right)}}\right) \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)} - 2$$
           1
-2 - -------------
2*csc(-2 + a)
$$-2 - \frac{1}{2 \csc{\left(a - 2 \right)}}$$
     cos(a)*sin(2)   cos(2)*sin(a)
-2 + ------------- - -------------
2               2      
$$- \frac{\sin{\left(a \right)} \cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)} \cos{\left(a \right)}}{2} - 2$$
               /    pi\             /    pi\
sin(2)*sin|a + --|   sin(a)*sin|2 + --|
\    2 /             \    2 /
-2 + ------------------ - ------------------
2                    2         
$$- \frac{\sin{\left(a \right)} \sin{\left(\frac{\pi}{2} + 2 \right)}}{2} + \frac{\sin{\left(2 \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{2} - 2$$
-2 + sin(2)*sin(a + pi/2)/2 - sin(a)*sin(2 + pi/2)/2
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