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Другие калькуляторы

(10*cos(a)-2*sin(a)+10)/(sin(a)-5*cos(a)+5) если a=2

Выражение, которое надо упростить:

Решение

Вы ввели [src]
10*cos(a) - 2*sin(a) + 10
-------------------------
  sin(a) - 5*cos(a) + 5  
$$\frac{- 2 \sin{\left(a \right)} + 10 \cos{\left(a \right)} + 10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
(10*cos(a) - 2*sin(a) + 10)/(sin(a) - 5*cos(a) + 5)
Общее упрощение [src]
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)  
$$\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
2*(5 - sin(a) + 5*cos(a))/(5 - 5*cos(a) + sin(a))
Подстановка условия [src]
(10*cos(a) - 2*sin(a) + 10)/(sin(a) - 5*cos(a) + 5) при a = 2
подставляем
10*cos(a) - 2*sin(a) + 10
-------------------------
  sin(a) - 5*cos(a) + 5  
$$\frac{- 2 \sin{\left(a \right)} + 10 \cos{\left(a \right)} + 10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)  
$$\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
переменные
a = 2
$$a = 2$$
2*(5 - sin((2)) + 5*cos((2)))
-----------------------------
  5 - 5*cos((2)) + sin((2))  
$$\frac{2 \left(- \sin{\left((2) \right)} + 5 \cos{\left((2) \right)} + 5\right)}{\sin{\left((2) \right)} - 5 \cos{\left((2) \right)} + 5}$$
2*(5 - sin(2) + 5*cos(2))
-------------------------
  5 - 5*cos(2) + sin(2)  
$$\frac{2 \cdot \left(5 \cos{\left(2 \right)} - \sin{\left(2 \right)} + 5\right)}{\sin{\left(2 \right)} - 5 \cos{\left(2 \right)} + 5}$$
2*(5 - sin(2) + 5*cos(2))/(5 - 5*cos(2) + sin(2))
Рациональный знаменатель [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
$$- \frac{2 \sin{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10 \cos{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
10/(5 - 5*cos(a) + sin(a)) - 2*sin(a)/(5 - 5*cos(a) + sin(a)) + 10*cos(a)/(5 - 5*cos(a) + sin(a))
Собрать выражение [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
$$- \frac{2 \sin{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10 \cos{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
10/(5 - 5*cos(a) + sin(a)) - 2*sin(a)/(5 - 5*cos(a) + sin(a)) + 10*cos(a)/(5 - 5*cos(a) + sin(a))
Раскрыть выражение [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
$$- \frac{2 \sin{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10 \cos{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
10/(5 - 5*cos(a) + sin(a)) - 2*sin(a)/(5 - 5*cos(a) + sin(a)) + 10*cos(a)/(5 - 5*cos(a) + sin(a))
Объединение рациональных выражений [src]
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)  
$$\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}$$
2*(5 - sin(a) + 5*cos(a))/(5 - 5*cos(a) + sin(a))
Комбинаторика [src]
-2*(5 - sin(a) + 5*cos(a))
--------------------------
  -5 - sin(a) + 5*cos(a)  
$$- \frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{- \sin{\left(a \right)} + 5 \cos{\left(a \right)} - 5}$$
-2*(5 - sin(a) + 5*cos(a))/(-5 - sin(a) + 5*cos(a))
Степени [src]
        I*a      -I*a     /   -I*a    I*a\
10 + 5*e    + 5*e     + I*\- e     + e   /
------------------------------------------
       I*a      -I*a     /   -I*a    I*a\ 
    5*e      5*e       I*\- e     + e   / 
5 - ------ - ------- - ------------------ 
      2         2              2          
$$\frac{i \left(e^{i a} - e^{- i a}\right) + 5 e^{i a} + 10 + 5 e^{- i a}}{- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} - \frac{5 e^{i a}}{2} + 5 - \frac{5 e^{- i a}}{2}}$$
(10 + 5*exp(i*a) + 5*exp(-i*a) + i*(-exp(-i*a) + exp(i*a)))/(5 - 5*exp(i*a)/2 - 5*exp(-i*a)/2 - i*(-exp(-i*a) + exp(i*a))/2)
Тригонометрическая часть [src]
2*(-5 - 5*cos(a) + sin(a))
--------------------------
  -5 - sin(a) + 5*cos(a)  
$$\frac{2 \left(\sin{\left(a \right)} - 5 \cos{\left(a \right)} - 5\right)}{- \sin{\left(a \right)} + 5 \cos{\left(a \right)} - 5}$$
          /    pi\            
10 - 2*cos|a - --| + 10*cos(a)
          \    2 /            
------------------------------
                    /    pi\  
  5 - 5*cos(a) + cos|a - --|  
                    \    2 /  
$$\frac{10 \cos{\left(a \right)} - 2 \cos{\left(a - \frac{\pi}{2} \right)} + 10}{- 5 \cos{\left(a \right)} + \cos{\left(a - \frac{\pi}{2} \right)} + 5}$$
       2        10  
10 - ------ + ------
     csc(a)   sec(a)
--------------------
      1        5    
5 + ------ - ------ 
    csc(a)   sec(a) 
$$\frac{10 + \frac{10}{\sec{\left(a \right)}} - \frac{2}{\csc{\left(a \right)}}}{5 - \frac{5}{\sec{\left(a \right)}} + \frac{1}{\csc{\left(a \right)}}}$$
                      /    pi\
10 - 2*sin(a) + 10*sin|a + --|
                      \    2 /
------------------------------
           /    pi\           
  5 - 5*sin|a + --| + sin(a)  
           \    2 /           
$$\frac{- 2 \sin{\left(a \right)} + 10 \sin{\left(a + \frac{\pi}{2} \right)} + 10}{\sin{\left(a \right)} - 5 \sin{\left(a + \frac{\pi}{2} \right)} + 5}$$
          2          10  
10 - ----------- + ------
        /    pi\   sec(a)
     sec|a - --|         
        \    2 /         
-------------------------
          1          5   
 5 + ----------- - ------
        /    pi\   sec(a)
     sec|a - --|         
        \    2 /         
$$\frac{10 - \frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{10}{\sec{\left(a \right)}}}{5 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}$$
       2           10    
10 - ------ + -----------
     csc(a)      /pi    \
              csc|-- - a|
                 \2     /
-------------------------
       1           5     
 5 + ------ - -----------
     csc(a)      /pi    \
              csc|-- - a|
                 \2     /
$$\frac{10 + \frac{10}{\csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(a \right)}}}{5 - \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(a \right)}}}$$
          2          10  
10 - ----------- + ------
        /pi    \   sec(a)
     sec|-- - a|         
        \2     /         
-------------------------
          1          5   
 5 + ----------- - ------
        /pi    \   sec(a)
     sec|-- - a|         
        \2     /         
$$\frac{10 - \frac{2}{\sec{\left(- a + \frac{\pi}{2} \right)}} + \frac{10}{\sec{\left(a \right)}}}{5 + \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}$$
          2             10    
10 - ----------- + -----------
     csc(pi - a)      /pi    \
                   csc|-- - a|
                      \2     /
------------------------------
         1             5      
5 + ----------- - ----------- 
    csc(pi - a)      /pi    \ 
                  csc|-- - a| 
                     \2     / 
$$\frac{10 + \frac{10}{\csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(- a + \pi \right)}}}{5 - \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- a + \pi \right)}}}$$
                                   /a\
10 + 10*cos(a) - 2*(1 + cos(a))*tan|-|
                                   \2/
--------------------------------------
                                 /a\  
  5 - 5*cos(a) + (1 + cos(a))*tan|-|  
                                 \2/  
$$\frac{- 2 \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 10 \cos{\left(a \right)} + 10}{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} - 5 \cos{\left(a \right)} + 5}$$
                 /       2/a   pi\\             
10 + 10*cos(a) - |1 - cot |- + --||*(1 + sin(a))
                 \        \2   4 //             
------------------------------------------------
                /       2/a   pi\\              
                |1 - cot |- + --||*(1 + sin(a)) 
                \        \2   4 //              
 5 - 5*cos(a) + ------------------------------- 
                               2                
$$\frac{- \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) + 10 \cos{\left(a \right)} + 10}{\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} - 5 \cos{\left(a \right)} + 5}$$
            /a\       /       2/a\\
       4*tan|-|    10*|1 - tan |-||
            \2/       \        \2//
10 - ----------- + ----------------
            2/a\            2/a\   
     1 + tan |-|     1 + tan |-|   
             \2/             \2/   
-----------------------------------
       /       2/a\\          /a\  
     5*|1 - tan |-||     2*tan|-|  
       \        \2//          \2/  
 5 - --------------- + ----------- 
              2/a\            2/a\ 
       1 + tan |-|     1 + tan |-| 
               \2/             \2/ 
$$\frac{\frac{10 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 10 - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{- \frac{5 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 5 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}$$
            /a\           /a   pi\ 
       4*tan|-|     20*tan|- + --| 
            \2/           \2   4 / 
10 - ----------- + ----------------
            2/a\          2/a   pi\
     1 + tan |-|   1 + tan |- + --|
             \2/           \2   4 /
-----------------------------------
            /a   pi\           /a\ 
      10*tan|- + --|      2*tan|-| 
            \2   4 /           \2/ 
 5 - ---------------- + -----------
            2/a   pi\          2/a\
     1 + tan |- + --|   1 + tan |-|
             \2   4 /           \2/
$$\frac{10 + \frac{20 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{5 - \frac{10 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}$$
            /a\           /a   pi\ 
       4*cot|-|     20*tan|- + --| 
            \2/           \2   4 / 
10 - ----------- + ----------------
            2/a\          2/a   pi\
     1 + cot |-|   1 + tan |- + --|
             \2/           \2   4 /
-----------------------------------
            /a   pi\           /a\ 
      10*tan|- + --|      2*cot|-| 
            \2   4 /           \2/ 
 5 - ---------------- + -----------
            2/a   pi\          2/a\
     1 + tan |- + --|   1 + cot |-|
             \2   4 /           \2/
$$\frac{10 - \frac{4 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{20 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}{5 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{10 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}$$
       /        2/a   pi\\      /        2/a\\
     2*|-1 + tan |- + --||   10*|-1 + cot |-||
       \         \2   4 //      \         \2//
10 - --------------------- + -----------------
               2/a   pi\               2/a\   
        1 + tan |- + --|        1 + cot |-|   
                \2   4 /                \2/   
----------------------------------------------
               2/a   pi\     /        2/a\\   
       -1 + tan |- + --|   5*|-1 + cot |-||   
                \2   4 /     \         \2//   
   5 + ----------------- - ----------------   
               2/a   pi\            2/a\      
        1 + tan |- + --|     1 + cot |-|      
                \2   4 /             \2/      
$$\frac{- \frac{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{10 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 10}{\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{5 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 5}$$
                               /       1   \
                            10*|1 - -------|
                               |       2/a\|
                               |    cot |-||
              4                \        \2//
10 - -------------------- + ----------------
     /       1   \    /a\            1      
     |1 + -------|*cot|-|     1 + -------   
     |       2/a\|    \2/            2/a\   
     |    cot |-||                cot |-|   
     \        \2//                    \2/   
--------------------------------------------
       /       1   \                        
     5*|1 - -------|                        
       |       2/a\|                        
       |    cot |-||                        
       \        \2//            2           
 5 - --------------- + -------------------- 
              1        /       1   \    /a\ 
       1 + -------     |1 + -------|*cot|-| 
              2/a\     |       2/a\|    \2/ 
           cot |-|     |    cot |-||        
               \2/     \        \2//        
$$\frac{\frac{10 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 10 - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 5 + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}$$
       /       2/a   pi\\      /       2/a\\
     2*|1 - cot |- + --||   10*|1 - tan |-||
       \        \2   4 //      \        \2//
10 - -------------------- + ----------------
              2/a   pi\              2/a\   
       1 + cot |- + --|       1 + tan |-|   
               \2   4 /               \2/   
--------------------------------------------
              2/a   pi\     /       2/a\\   
       1 - cot |- + --|   5*|1 - tan |-||   
               \2   4 /     \        \2//   
   5 + ---------------- - ---------------   
              2/a   pi\            2/a\     
       1 + cot |- + --|     1 + tan |-|     
               \2   4 /             \2/     
$$\frac{\frac{10 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 10}{- \frac{5 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + 5}$$
       //  0     for a mod pi = 0\      //  1     for a mod 2*pi = 0\
10 - 2*|<                        | + 10*|<                          |
       \\sin(a)     otherwise    /      \\cos(a)      otherwise     /
---------------------------------------------------------------------
        //  1     for a mod 2*pi = 0\   //  0     for a mod pi = 0\  
  5 - 5*|<                          | + |<                        |  
        \\cos(a)      otherwise     /   \\sin(a)     otherwise    /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                        //     1       for a mod 2*pi = 0\
       //  0     for a mod pi = 0\      ||                               |
10 - 2*|<                        | + 10*|<   /    pi\                    |
       \\sin(a)     otherwise    /      ||sin|a + --|      otherwise     |
                                        \\   \    2 /                    /
--------------------------------------------------------------------------
        //     1       for a mod 2*pi = 0\                                
        ||                               |   //  0     for a mod pi = 0\  
  5 - 5*|<   /    pi\                    | + |<                        |  
        ||sin|a + --|      otherwise     |   \\sin(a)     otherwise    /  
        \\   \    2 /                    /                                
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //     0       for a mod pi = 0\                                   
       ||                             |      //  1     for a mod 2*pi = 0\
10 - 2*|<   /    pi\                  | + 10*|<                          |
       ||cos|a - --|     otherwise    |      \\cos(a)      otherwise     /
       \\   \    2 /                  /                                   
--------------------------------------------------------------------------
                                        //     0       for a mod pi = 0\  
        //  1     for a mod 2*pi = 0\   ||                             |  
  5 - 5*|<                          | + |<   /    pi\                  |  
        \\cos(a)      otherwise     /   ||cos|a - --|     otherwise    |  
                                        \\   \    2 /                  /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //            /    3*pi\             \                                   
       ||  1     for |a + ----| mod 2*pi = 0|      //  1     for a mod 2*pi = 0\
10 - 2*|<            \     2  /             | + 10*|<                          |
       ||                                   |      \\cos(a)      otherwise     /
       \\sin(a)           otherwise         /                                   
--------------------------------------------------------------------------------
                                        //            /    3*pi\             \  
        //  1     for a mod 2*pi = 0\   ||  1     for |a + ----| mod 2*pi = 0|  
  5 - 5*|<                          | + |<            \     2  /             |  
        \\cos(a)      otherwise     /   ||                                   |  
                                        \\sin(a)           otherwise         /  
$$\frac{\left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 5}$$
       //     0       for a mod pi = 0\                                   
       ||                             |      //  1     for a mod 2*pi = 0\
       ||     1                       |      ||                          |
10 - 2*|<-----------     otherwise    | + 10*|<  1                       |
       ||   /    pi\                  |      ||------      otherwise     |
       ||sec|a - --|                  |      \\sec(a)                    /
       \\   \    2 /                  /                                   
--------------------------------------------------------------------------
                                        //     0       for a mod pi = 0\  
        //  1     for a mod 2*pi = 0\   ||                             |  
        ||                          |   ||     1                       |  
  5 - 5*|<  1                       | + |<-----------     otherwise    |  
        ||------      otherwise     |   ||   /    pi\                  |  
        \\sec(a)                    /   ||sec|a - --|                  |  
                                        \\   \    2 /                  /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                        //     1       for a mod 2*pi = 0\
       //  0     for a mod pi = 0\      ||                               |
       ||                        |      ||     1                         |
10 - 2*|<  1                     | + 10*|<-----------      otherwise     |
       ||------     otherwise    |      ||   /pi    \                    |
       \\csc(a)                  /      ||csc|-- - a|                    |
                                        \\   \2     /                    /
--------------------------------------------------------------------------
        //     1       for a mod 2*pi = 0\                                
        ||                               |   //  0     for a mod pi = 0\  
        ||     1                         |   ||                        |  
  5 - 5*|<-----------      otherwise     | + |<  1                     |  
        ||   /pi    \                    |   ||------     otherwise    |  
        ||csc|-- - a|                    |   \\csc(a)                  /  
        \\   \2     /                    /                                
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                             2/a\        
                                        8*sin |-|*sin(a) 
     20*(-1 - cos(2*a) + 2*cos(a))            \2/        
10 + ------------------------------ - -------------------
                                  2      2           4/a\
     1 - cos(2*a) + 2*(1 - cos(a))    sin (a) + 4*sin |-|
                                                      \2/
---------------------------------------------------------
                                             2/a\        
                                        4*sin |-|*sin(a) 
     10*(-1 - cos(2*a) + 2*cos(a))            \2/        
 5 - ------------------------------ + -------------------
                                  2      2           4/a\
     1 - cos(2*a) + 2*(1 - cos(a))    sin (a) + 4*sin |-|
                                                      \2/
$$\frac{- \frac{8 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + 10 + \frac{20 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1}}{\frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + 5 - \frac{10 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1}}$$
       //    0       for a mod pi = 0\                                   
       ||                            |                                   
       ||1 - cos(a)                  |      //  1     for a mod 2*pi = 0\
10 - 2*|<----------     otherwise    | + 10*|<                          |
       ||     /a\                    |      \\cos(a)      otherwise     /
       ||  tan|-|                    |                                   
       \\     \2/                    /                                   
-------------------------------------------------------------------------
                                        //    0       for a mod pi = 0\  
                                        ||                            |  
        //  1     for a mod 2*pi = 0\   ||1 - cos(a)                  |  
  5 - 5*|<                          | + |<----------     otherwise    |  
        \\cos(a)      otherwise     /   ||     /a\                    |  
                                        ||  tan|-|                    |  
                                        \\     \2/                    /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 5}$$
        /         4/a\\                         
        |    4*sin |-||                         
        |          \2/|                         
     10*|1 - ---------|              2/a\       
        |        2    |         8*sin |-|       
        \     sin (a) /               \2/       
10 + ------------------ - ----------------------
                4/a\      /         4/a\\       
           4*sin |-|      |    4*sin |-||       
                 \2/      |          \2/|       
       1 + ---------      |1 + ---------|*sin(a)
               2          |        2    |       
            sin (a)       \     sin (a) /       
------------------------------------------------
       /         4/a\\                          
       |    4*sin |-||                          
       |          \2/|                          
     5*|1 - ---------|              2/a\        
       |        2    |         4*sin |-|        
       \     sin (a) /               \2/        
 5 - ----------------- + ---------------------- 
                4/a\     /         4/a\\        
           4*sin |-|     |    4*sin |-||        
                 \2/     |          \2/|        
       1 + ---------     |1 + ---------|*sin(a) 
               2         |        2    |        
            sin (a)      \     sin (a) /        
$$\frac{\frac{10 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + 10 - \frac{8 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}{- \frac{5 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + 5 + \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}$$
                                        //                              /    pi\           \
                                        ||           0              for |a + --| mod pi = 0|
       //  0     for a mod pi = 0\      ||                              \    2 /           |
10 - 2*|<                        | + 10*|<                                                 |
       \\sin(a)     otherwise    /      ||                /a   pi\                         |
                                        ||(1 + sin(a))*cot|- + --|         otherwise       |
                                        \\                \2   4 /                         /
--------------------------------------------------------------------------------------------
        //                              /    pi\           \                                
        ||           0              for |a + --| mod pi = 0|                                
        ||                              \    2 /           |   //  0     for a mod pi = 0\  
  5 - 5*|<                                                 | + |<                        |  
        ||                /a   pi\                         |   \\sin(a)     otherwise    /  
        ||(1 + sin(a))*cot|- + --|         otherwise       |                                
        \\                \2   4 /                         /                                
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //     0       for a mod pi = 0\      //     1        for a mod 2*pi = 0\
       ||                             |      ||                                |
       ||       /a\                   |      ||        2/a\                    |
       ||  2*cot|-|                   |      ||-1 + cot |-|                    |
10 - 2*|<       \2/                   | + 10*|<         \2/                    |
       ||-----------     otherwise    |      ||------------      otherwise     |
       ||       2/a\                  |      ||       2/a\                     |
       ||1 + cot |-|                  |      ||1 + cot |-|                     |
       \\        \2/                  /      \\        \2/                     /
--------------------------------------------------------------------------------
        //     1        for a mod 2*pi = 0\   //     0       for a mod pi = 0\  
        ||                                |   ||                             |  
        ||        2/a\                    |   ||       /a\                   |  
        ||-1 + cot |-|                    |   ||  2*cot|-|                   |  
  5 - 5*|<         \2/                    | + |<       \2/                   |  
        ||------------      otherwise     |   ||-----------     otherwise    |  
        ||       2/a\                     |   ||       2/a\                  |  
        ||1 + cot |-|                     |   ||1 + cot |-|                  |  
        \\        \2/                     /   \\        \2/                  /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //     0       for a mod pi = 0\      //     1       for a mod 2*pi = 0\
       ||                             |      ||                               |
       ||       /a\                   |      ||       2/a\                    |
       ||  2*tan|-|                   |      ||1 - tan |-|                    |
10 - 2*|<       \2/                   | + 10*|<        \2/                    |
       ||-----------     otherwise    |      ||-----------      otherwise     |
       ||       2/a\                  |      ||       2/a\                    |
       ||1 + tan |-|                  |      ||1 + tan |-|                    |
       \\        \2/                  /      \\        \2/                    /
-------------------------------------------------------------------------------
        //     1       for a mod 2*pi = 0\   //     0       for a mod pi = 0\  
        ||                               |   ||                             |  
        ||       2/a\                    |   ||       /a\                   |  
        ||1 - tan |-|                    |   ||  2*tan|-|                   |  
  5 - 5*|<        \2/                    | + |<       \2/                   |  
        ||-----------      otherwise     |   ||-----------     otherwise    |  
        ||       2/a\                    |   ||       2/a\                  |  
        ||1 + tan |-|                    |   ||1 + tan |-|                  |  
        \\        \2/                    /   \\        \2/                  /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //            0              for a mod pi = 0\      //             1               for a mod 2*pi = 0\
       ||                                           |      ||                                               |
10 - 2*|
            
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                                      //     1        for a mod 2*pi = 0\
                                                      ||                                |
       //         0            for a mod pi = 0\      ||        1                       |
       ||                                      |      ||-1 + -------                    |
       ||         2                            |      ||        2/a\                    |
       ||--------------------     otherwise    |      ||     tan |-|                    |
10 - 2*|
            
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
        /         2/a\   \                                 
        |      sec |-|   |                                 
        |          \2/   |                                 
     10*|1 - ------------|                                 
        |       2/a   pi\|                   /a\           
        |    sec |- - --||              4*sec|-|           
        \        \2   2 //                   \2/           
10 + --------------------- - ------------------------------
                 2/a\        /         2/a\   \            
              sec |-|        |      sec |-|   |            
                  \2/        |          \2/   |    /a   pi\
        1 + ------------     |1 + ------------|*sec|- - --|
               2/a   pi\     |       2/a   pi\|    \2   2 /
            sec |- - --|     |    sec |- - --||            
                \2   2 /     \        \2   2 //            
-----------------------------------------------------------
       /         2/a\   \                                  
       |      sec |-|   |                                  
       |          \2/   |                                  
     5*|1 - ------------|                                  
       |       2/a   pi\|                   /a\            
       |    sec |- - --||              2*sec|-|            
       \        \2   2 //                   \2/            
 5 - -------------------- + ------------------------------ 
                2/a\        /         2/a\   \             
             sec |-|        |      sec |-|   |             
                 \2/        |          \2/   |    /a   pi\ 
       1 + ------------     |1 + ------------|*sec|- - --| 
              2/a   pi\     |       2/a   pi\|    \2   2 / 
           sec |- - --|     |    sec |- - --||             
               \2   2 /     \        \2   2 //             
$$\frac{\frac{10 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + 10 - \frac{4 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{- \frac{5 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + 5 + \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}$$
        /       2/a   pi\\                            
        |    cos |- - --||                            
        |        \2   2 /|                            
     10*|1 - ------------|                            
        |         2/a\   |              /a   pi\      
        |      cos |-|   |         4*cos|- - --|      
        \          \2/   /              \2   2 /      
10 + --------------------- - -------------------------
               2/a   pi\     /       2/a   pi\\       
            cos |- - --|     |    cos |- - --||       
                \2   2 /     |        \2   2 /|    /a\
        1 + ------------     |1 + ------------|*cos|-|
                 2/a\        |         2/a\   |    \2/
              cos |-|        |      cos |-|   |       
                  \2/        \          \2/   /       
------------------------------------------------------
       /       2/a   pi\\                             
       |    cos |- - --||                             
       |        \2   2 /|                             
     5*|1 - ------------|                             
       |         2/a\   |              /a   pi\       
       |      cos |-|   |         2*cos|- - --|       
       \          \2/   /              \2   2 /       
 5 - -------------------- + ------------------------- 
              2/a   pi\     /       2/a   pi\\        
           cos |- - --|     |    cos |- - --||        
               \2   2 /     |        \2   2 /|    /a\ 
       1 + ------------     |1 + ------------|*cos|-| 
                2/a\        |         2/a\   |    \2/ 
             cos |-|        |      cos |-|   |        
                 \2/        \          \2/   /        
$$\frac{\frac{10 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + 10 - \frac{4 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + 5 + \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}$$
                                             //                      /    pi\           \
       //     0       for a mod pi = 0\      ||       0          for |a + --| mod pi = 0|
       ||                             |      ||                      \    2 /           |
       ||       /a\                   |      ||                                         |
       ||  2*cot|-|                   |      ||      /a   pi\                           |
10 - 2*|<       \2/                   | + 10*|< 2*cot|- + --|                           |
       ||-----------     otherwise    |      ||      \2   4 /                           |
       ||       2/a\                  |      ||----------------         otherwise       |
       ||1 + cot |-|                  |      ||       2/a   pi\                         |
       \\        \2/                  /      ||1 + cot |- + --|                         |
                                             \\        \2   4 /                         /
-----------------------------------------------------------------------------------------
        //                      /    pi\           \                                     
        ||       0          for |a + --| mod pi = 0|   //     0       for a mod pi = 0\  
        ||                      \    2 /           |   ||                             |  
        ||                                         |   ||       /a\                   |  
        ||      /a   pi\                           |   ||  2*cot|-|                   |  
  5 - 5*|< 2*cot|- + --|                           | + |<       \2/                   |  
        ||      \2   4 /                           |   ||-----------     otherwise    |  
        ||----------------         otherwise       |   ||       2/a\                  |  
        ||       2/a   pi\                         |   ||1 + cot |-|                  |  
        ||1 + cot |- + --|                         |   \\        \2/                  /  
        \\        \2   4 /                         /                                     
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}$$
        /       2/pi   a\\                            
        |    csc |-- - -||                            
        |        \2    2/|                            
     10*|1 - ------------|                            
        |         2/a\   |              /pi   a\      
        |      csc |-|   |         4*csc|-- - -|      
        \          \2/   /              \2    2/      
10 + --------------------- - -------------------------
               2/pi   a\     /       2/pi   a\\       
            csc |-- - -|     |    csc |-- - -||       
                \2    2/     |        \2    2/|    /a\
        1 + ------------     |1 + ------------|*csc|-|
                 2/a\        |         2/a\   |    \2/
              csc |-|        |      csc |-|   |       
                  \2/        \          \2/   /       
------------------------------------------------------
       /       2/pi   a\\                             
       |    csc |-- - -||                             
       |        \2    2/|                             
     5*|1 - ------------|                             
       |         2/a\   |              /pi   a\       
       |      csc |-|   |         2*csc|-- - -|       
       \          \2/   /              \2    2/       
 5 - -------------------- + ------------------------- 
              2/pi   a\     /       2/pi   a\\        
           csc |-- - -|     |    csc |-- - -||        
               \2    2/     |        \2    2/|    /a\ 
       1 + ------------     |1 + ------------|*csc|-| 
                2/a\        |         2/a\   |    \2/ 
             csc |-|        |      csc |-|   |        
                 \2/        \          \2/   /        
$$\frac{\frac{10 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + 10 - \frac{4 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + 5 + \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}$$
       //                       /    3*pi\             \                                         
       ||        1          for |a + ----| mod 2*pi = 0|      //     1        for a mod 2*pi = 0\
       ||                       \     2  /             |      ||                                |
       ||                                              |      ||        2/a\                    |
       ||        2/a   pi\                             |      ||-1 + cot |-|                    |
10 - 2*|<-1 + tan |- + --|                             | + 10*|<         \2/                    |
       ||         \2   4 /                             |      ||------------      otherwise     |
       ||-----------------           otherwise         |      ||       2/a\                     |
       ||        2/a   pi\                             |      ||1 + cot |-|                     |
       || 1 + tan |- + --|                             |      \\        \2/                     /
       \\         \2   4 /                             /                                         
-------------------------------------------------------------------------------------------------
                                              //                       /    3*pi\             \  
        //     1        for a mod 2*pi = 0\   ||        1          for |a + ----| mod 2*pi = 0|  
        ||                                |   ||                       \     2  /             |  
        ||        2/a\                    |   ||                                              |  
        ||-1 + cot |-|                    |   ||        2/a   pi\                             |  
  5 - 5*|<         \2/                    | + |<-1 + tan |- + --|                             |  
        ||------------      otherwise     |   ||         \2   4 /                             |  
        ||       2/a\                     |   ||-----------------           otherwise         |  
        ||1 + cot |-|                     |   ||        2/a   pi\                             |  
        \\        \2/                     /   || 1 + tan |- + --|                             |  
                                              \\         \2   4 /                             /  
$$\frac{\left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 5}$$
       //             0                for a mod pi = 0\                                                        
       ||                                              |                                                        
       ||          2*sin(a)                            |      //             1               for a mod 2*pi = 0\
       ||----------------------------     otherwise    |      ||                                               |
       ||             /        2    \                  |      ||           2                                   |
10 - 2*|<             |     sin (a) |                  | + 10*|< -4 + 4*sin (a) + 4*cos(a)                     |
       ||(1 - cos(a))*|1 + ---------|                  |      ||---------------------------      otherwise     |
       ||             |         4/a\|                  |      ||              2        2                       |
       ||             |    4*sin |-||                  |      \\2*(1 - cos(a))  + 2*sin (a)                    /
       ||             \          \2//                  |                                                        
       \\                                              /                                                        
----------------------------------------------------------------------------------------------------------------
                                                             //             0                for a mod pi = 0\  
                                                             ||                                              |  
        //             1               for a mod 2*pi = 0\   ||          2*sin(a)                            |  
        ||                                               |   ||----------------------------     otherwise    |  
        ||           2                                   |   ||             /        2    \                  |  
  5 - 5*|< -4 + 4*sin (a) + 4*cos(a)                     | + |<             |     sin (a) |                  |  
        ||---------------------------      otherwise     |   ||(1 - cos(a))*|1 + ---------|                  |  
        ||              2        2                       |   ||             |         4/a\|                  |  
        \\2*(1 - cos(a))  + 2*sin (a)                    /   ||             |    4*sin |-||                  |  
                                                             ||             \          \2//                  |  
                                                             \\                                              /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                                         //      1         for a mod 2*pi = 0\
                                                         ||                                  |
       //           0             for a mod pi = 0\      ||         2                        |
       ||                                         |      ||      sin (a)                     |
       ||         sin(a)                          |      ||-1 + ---------                    |
       ||-----------------------     otherwise    |      ||          4/a\                    |
       ||/        2    \                          |      ||     4*sin |-|                    |
10 - 2*|<|     sin (a) |    2/a\                  | + 10*|<           \2/                    |
       |||1 + ---------|*sin |-|                  |      ||--------------      otherwise     |
       |||         4/a\|     \2/                  |      ||        2                         |
       |||    4*sin |-||                          |      ||     sin (a)                      |
       ||\          \2//                          |      ||1 + ---------                     |
       \\                                         /      ||         4/a\                     |
                                                         ||    4*sin |-|                     |
                                                         \\          \2/                     /
----------------------------------------------------------------------------------------------
        //      1         for a mod 2*pi = 0\                                                 
        ||                                  |                                                 
        ||         2                        |   //           0             for a mod pi = 0\  
        ||      sin (a)                     |   ||                                         |  
        ||-1 + ---------                    |   ||         sin(a)                          |  
        ||          4/a\                    |   ||-----------------------     otherwise    |  
        ||     4*sin |-|                    |   ||/        2    \                          |  
  5 - 5*|<           \2/                    | + |<|     sin (a) |    2/a\                  |  
        ||--------------      otherwise     |   |||1 + ---------|*sin |-|                  |  
        ||        2                         |   |||         4/a\|     \2/                  |  
        ||     sin (a)                      |   |||    4*sin |-||                          |  
        ||1 + ---------                     |   ||\          \2//                          |  
        ||         4/a\                     |   \\                                         /  
        ||    4*sin |-|                     |                                                 
        \\          \2/                     /                                                 
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
       //              0                 for a mod pi = 0\      //                1                  for a mod 2*pi = 0\
       ||                                                |      ||                                                     |
       ||/     0       for a mod pi = 0                  |      ||/     1        for a mod 2*pi = 0                    |
       |||                                               |      |||                                                    |
       |||       /a\                                     |      |||        2/a\                                        |
10 - 2*|<|  2*cot|-|                                     | + 10*|<|-1 + cot |-|                                        |
       ||<       \2/                        otherwise    |      ||<         \2/                          otherwise     |
       |||-----------     otherwise                      |      |||------------      otherwise                         |
       |||       2/a\                                    |      |||       2/a\                                         |
       |||1 + cot |-|                                    |      |||1 + cot |-|                                         |
       \\\        \2/                                    /      \\\        \2/                                         /
------------------------------------------------------------------------------------------------------------------------
        //                1                  for a mod 2*pi = 0\   //              0                 for a mod pi = 0\  
        ||                                                     |   ||                                                |  
        ||/     1        for a mod 2*pi = 0                    |   ||/     0       for a mod pi = 0                  |  
        |||                                                    |   |||                                               |  
        |||        2/a\                                        |   |||       /a\                                     |  
  5 - 5*|<|-1 + cot |-|                                        | + |<|  2*cot|-|                                     |  
        ||<         \2/                          otherwise     |   ||<       \2/                        otherwise    |  
        |||------------      otherwise                         |   |||-----------     otherwise                      |  
        |||       2/a\                                         |   |||       2/a\                                    |  
        |||1 + cot |-|                                         |   |||1 + cot |-|                                    |  
        \\\        \2/                                         /   \\\        \2/                                    /  
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                                                //        1          for a mod 2*pi = 0\
                                                                ||                                     |
       //              0                 for a mod pi = 0\      ||          2/a\                       |
       ||                                                |      ||       cos |-|                       |
       ||                /a\                             |      ||           \2/                       |
       ||           2*cos|-|                             |      ||-1 + ------------                    |
       ||                \2/                             |      ||        2/a   pi\                    |
       ||------------------------------     otherwise    |      ||     cos |- - --|                    |
10 - 2*|
            
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                                           //        1          for a mod 2*pi = 0\
                                                           ||                                     |
       //            0              for a mod pi = 0\      ||        2/a   pi\                    |
       ||                                           |      ||     sec |- - --|                    |
       ||           /a   pi\                        |      ||         \2   2 /                    |
       ||      2*sec|- - --|                        |      ||-1 + ------------                    |
       ||           \2   2 /                        |      ||          2/a\                       |
       ||-------------------------     otherwise    |      ||       sec |-|                       |
10 - 2*|
            
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) + 5}$$
                                                                //        1          for a mod 2*pi = 0\
                                                                ||                                     |
       //              0                 for a mod pi = 0\      ||          2/a\                       |
       ||                                                |      ||       csc |-|                       |
       ||                /a\                             |      ||           \2/                       |
       ||           2*csc|-|                             |      ||-1 + ------------                    |
       ||                \2/                             |      ||        2/pi   a\                    |
       ||------------------------------     otherwise    |      ||     csc |-- - -|                    |
10 - 2*|
            
$$\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}$$
(10 - 2*Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True)) + 10*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True)))/(5 - 5*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True)) + Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True)))
Общий знаменатель [src]
               20          
-2 - ----------------------
     -5 - sin(a) + 5*cos(a)
$$-2 - \frac{20}{- \sin{\left(a \right)} + 5 \cos{\left(a \right)} - 5}$$
-2 - 20/(-5 - sin(a) + 5*cos(a))
Численный ответ [src]
(10.0 + 10.0*cos(a) - 2.0*sin(a))/(5.0 - 5.0*cos(a) + sin(a))
(10.0 + 10.0*cos(a) - 2.0*sin(a))/(5.0 - 5.0*cos(a) + sin(a))