sin(3*pi/2-a)-cos(pi+a)еслиa=4 (упростите выражение)

Вы ввели

   /3*pi    \              
sin|---- - a| - cos(pi + a)
   \ 2      /              

$$\sin{\left(- a + \frac{3 \pi}{2} \right)} - \cos{\left(a + \pi \right)}$$

Подстановка условия

sin((3*pi)/2 - a) - cos(pi + a) при a = 4

sin((3*pi)/2 - a) - cos(pi + a)

$$\sin{\left (- a + \frac{3 \pi}{2} \right )} - \cos{\left (a + \pi \right )}$$

sin((3*pi)/2 - (4)) - cos(pi + (4))

$$\sin{\left (- (4) + \frac{3 \pi}{2} \right )} - \cos{\left ((4) + \pi \right )}$$

sin((3*pi)/2 - 4) - cos(pi + 4)

$$- \cos{\left (\pi + 4 \right )} + \sin{\left (- 4 + \frac{3 \pi}{2} \right )}$$

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Степени

                                 /     /    3*pi\      /     3*pi\\
                                 |   I*|a - ----|    I*|-a + ----||
   I*(pi + a)    I*(-pi - a)     |     \     2  /      \      2  /|
  e             e              I*\- e             + e             /
- ----------- - ------------ - ------------------------------------
       2             2                          2                  

$$- \frac{i \left(e^{i \left(- a + \frac{3 \pi}{2}\right)} - e^{i \left(a - \frac{3 \pi}{2}\right)}\right)}{2} - \frac{e^{i \left(- a - \pi\right)}}{2} - \frac{e^{i \left(a + \pi\right)}}{2}$$

0

$$0$$

-cos(a) - cos(pi + a)

$$- \cos{\left(a \right)} - \cos{\left(a + \pi \right)}$$

Численный ответ

-cos(pi + a) + sin(3*pi/2 - a)

Рациональный знаменатель

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$$0$$

Объединение рациональных выражений

            /-2*a + 3*pi\
cos(a) + sin|-----------|
            \     2     /

$$\sin{\left(\frac{- 2 a + 3 \pi}{2} \right)} + \cos{\left(a \right)}$$

Общее упрощение

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Собрать выражение

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Комбинаторика

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Общий знаменатель

-cos(a) - cos(pi + a)

$$- \cos{\left(a \right)} - \cos{\left(a + \pi \right)}$$

Тригонометрическая часть

                                                          //                                  /           /pi    \           \\
                                                          ||            0              for And|im(a) = 0, |-- - a| mod pi = 0||
  //   1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                                  \           \2     /           /|
- |<                                                  | + |<                                                                  |
  \\-cos(a)                  otherwise                /   ||                 /a   pi\                                         |
                                                          ||-(1 + sin(a))*cot|- + --|                 otherwise               |
                                                          \\                 \2   4 /                                         /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$

      //                        /           a             \\                                                      
      ||       1         for And|im(a) = 0, - mod 2*pi = 0||                                                      
      ||                        \           2             /|   //     1        for And(im(a) = 0, a mod 2*pi = 0)\
      ||                                                   |   ||                                                |
      ||              2                                    |   ||        2/a\                                    |
      ||/        2/a\\                                     |   ||-1 + cot |-|                                    |
1 - 2*|<|-1 + cot |-||                                     | + |<         \2/                                    |
      ||\         \4//                                     |   ||------------              otherwise             |
      ||---------------              otherwise             |   ||       2/a\                                     |
      ||              2                                    |   ||1 + cot |-|                                     |
      || /       2/a\\                                     |   \\        \2/                                     /
      || |1 + cot |-||                                     |                                                      
      \\ \        \4//                                     /                                                      

$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$

         2/pi   a\      /    pi\
1 - 2*sin |-- + -| + sin|a + --|
          \2    2/      \    2 /

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

         2/a\         
1 - 2*cos |-| + cos(a)
          \2/         

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

      1         2   
1 + ------ - -------
    sec(a)      2/a\
             sec |-|
                 \2/

$$1 + \frac{1}{\sec{\left(a \right)}} - \frac{2}{\sec^{2}{\left(\frac{a}{2} \right)}}$$

                                                          //                                        /           /pi    \           \\
                                                          ||               0                 for And|im(a) = 0, |-- - a| mod pi = 0||
                                                          ||                                        \           \2     /           /|
                                                          ||                                                                        |
  //   1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   || /       /    pi\\    /a   pi\                                          |
- |<                                                  | + |<-|1 + cos|a - --||*cos|- + --|                                          |
  \\-cos(a)                  otherwise                /   || \       \    2 //    \2   4 /                                          |
                                                          ||-------------------------------                 otherwise               |
                                                          ||             /a   pi\                                                   |
                                                          ||          cos|- - --|                                                   |
                                                          \\             \2   4 /                                                   /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$

         2/a\         /a   pi\  
  1 - cot |-|    2*tan|- + --|  
          \2/         \2   4 /  
- ----------- - ----------------
         2/a\          2/a   pi\
  1 + cot |-|   1 + tan |- + --|
          \2/           \2   4 /

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

                                                          //                /           /pi    \           \\
  //   1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||   0     for And|im(a) = 0, |-- - a| mod pi = 0||
- |<                                                  | + |<                \           \2     /           /|
  \\-cos(a)                  otherwise                /   ||                                                |
                                                          \\-cos(a)                 otherwise               /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$

0

$$0$$

         1                      
  1 - -------                   
         2/a\         /a   pi\  
      tan |-|    2*tan|- + --|  
          \2/         \2   4 /  
- ----------- - ----------------
         1             2/a   pi\
  1 + -------   1 + tan |- + --|
         2/a\           \2   4 /
      tan |-|                   
          \2/                   

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

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

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

                                                                  //                            /           /pi    \           \\
                                                                  ||         0           for And|im(a) = 0, |-- - a| mod pi = 0||
                                                                  ||                            \           \2     /           /|
  //       1         for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                                                            |
  ||                                                          |   || /           /a\ \                                          |
  || /       2/a\\                                            |   || |      2*tan|-| |                                          |
  ||-|1 - tan |-||                                            |   || |           \2/ |                                          |
- |< \        \2//                                            | + |<-|1 + -----------|                                          |
  ||---------------                  otherwise                |   || |           2/a\|                                          |
  ||         2/a\                                             |   || |    1 + tan |-||                                          |
  ||  1 + tan |-|                                             |   || \            \2//                                          |
  \\          \2/                                             /   ||-------------------                 otherwise               |
                                                                  ||       /a   pi\                                             |
                                                                  ||    tan|- + --|                                             |
                                                                  \\       \2   4 /                                             /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

                                                               //                         /           /pi    \           \\
  //     1        for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||       0          for And|im(a) = 0, |-- - a| mod pi = 0||
  ||                                                       |   ||                         \           \2     /           /|
  ||        2/a\                                           |   ||                                                         |
  ||-1 + tan |-|                                           |   ||       /a   pi\                                          |
- |<         \2/                                           | + |< -2*cot|- + --|                                          |
  ||------------                  otherwise                |   ||       \2   4 /                                          |
  ||       2/a\                                            |   ||----------------                 otherwise               |
  ||1 + tan |-|                                            |   ||       2/a   pi\                                         |
  \\        \2/                                            /   ||1 + cot |- + --|                                         |
                                                               \\        \2   4 /                                         /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$

cos(a) + cos(pi - a)

$$\cos{\left(a \right)} + \cos{\left(\pi - a \right)}$$

  1           1     
------ - -----------
sec(a)      /pi    \
         csc|-- - a|
            \2     /

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

  //     1        for And(im(a) = 0, (pi + a) mod 2*pi = 0)\                                                               
  ||                                                       |   //                         /           /pi    \           \\
  ||        1                                              |   ||       0          for And|im(a) = 0, |-- - a| mod pi = 0||
  ||-1 + -------                                           |   ||                         \           \2     /           /|
  ||        2/a\                                           |   ||                                                         |
  ||     cot |-|                                           |   ||       /a   pi\                                          |
- |<         \2/                                           | + |< -2*cot|- + --|                                          |
  ||------------                  otherwise                |   ||       \2   4 /                                          |
  ||       1                                               |   ||----------------                 otherwise               |
  ||1 + -------                                            |   ||       2/a   pi\                                         |
  ||       2/a\                                            |   ||1 + cot |- + --|                                         |
  ||    cot |-|                                            |   \\        \2   4 /                                         /
  \\        \2/                                            /                                                               

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$

         1             2      
1 + ----------- - ------------
       /pi    \      2/pi   a\
    csc|-- - a|   csc |-- - -|
       \2     /       \2    2/

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

     /    pi\         
- sin|a + --| + cos(a)
     \    2 /         

$$- \sin{\left(a + \frac{\pi}{2} \right)} + \cos{\left(a \right)}$$

                                                              //                                   /           /pi    \           \\
                                                              ||            0               for And|im(a) = 0, |-- - a| mod pi = 0||
  //     1       for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                                   \           \2     /           /|
  ||                                                      |   ||                                                                   |
  ||    -1                                                |   || /      1   \    /a   pi\                                          |
- |<-----------                  otherwise                | + |<-|1 + ------|*csc|- + --|                                          |
  ||   /pi    \                                           |   || \    csc(a)/    \2   4 /                                          |
  ||csc|-- - a|                                           |   ||--------------------------                 otherwise               |
  \\   \2     /                                           /   ||         /  a   pi\                                                |
                                                              ||      csc|- - + --|                                                |
                                                              \\         \  2   4 /                                                /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\left(1 + \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$

             /    3*pi\
-cos(a) - sin|a + ----|
             \     2  /

$$- \sin{\left(a + \frac{3 \pi}{2} \right)} - \cos{\left(a \right)}$$

     1          1   
----------- - ------
   /pi    \   sec(a)
csc|-- - a|         
   \2     /         

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

      //                   /           a             \\                                                
      ||    1       for And|im(a) = 0, - mod 2*pi = 0||                                                
      ||                   \           2             /|   //  1     for And(im(a) = 0, a mod 2*pi = 0)\
1 - 2*|<                                              | + |<                                          |
      ||1 + cos(a)                                    |   \\cos(a)              otherwise             /
      ||----------              otherwise             |                                                
      \\    2                                         /                                                

$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \frac{a}{2} \bmod 2 \pi = 0 \\\frac{\cos{\left(a \right)} + 1}{2} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1$$

                                                               //                              /           /pi    \           \\
                                                               ||          0            for And|im(a) = 0, |-- - a| mod pi = 0||
  //     1        for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                              \           \2     /           /|
  ||                                                       |   ||                                                              |
- |<    /    pi\                                           | + |<-(1 + sin(a))*cos(a)                                          |
  ||-sin|a + --|                  otherwise                |   ||---------------------                 otherwise               |
  \\    \    2 /                                           /   ||         2/a   pi\                                            |
                                                               ||    2*sin |- + --|                                            |
                                                               \\          \2   4 /                                            /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(a \right)}}{2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$

                                                                                                       //                                                                            /           /pi    \           \\
  //                         1                            for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                                 0                                   for And|im(a) = 0, |-- - a| mod pi = 0||
  ||                                                                                               |   ||                                                                            \           \2     /           /|
  || //     1        for And(im(a) = 0, a mod 2*pi = 0)\                                           |   ||                                                                                                            |
  || ||                                                |                                           |   || /    //     0       for And(im(a) = 0, a mod pi = 0)\\                                                     |
  || ||        2/a\                                    |                                           |   || |    ||                                             ||                                                     |
- |< ||-1 + cot |-|                                    |                                           | + |< |    ||       /a\                                   ||                                                     |
  ||-|<         \2/                                    |                  otherwise                |   || |    ||  2*cot|-|                                   ||    /a   pi\                                         |
  || ||------------              otherwise             |                                           |   ||-|1 + |<       \2/                                   ||*cot|- + --|                 otherwise               |
  || ||       2/a\                                     |                                           |   || |    ||-----------             otherwise            ||    \2   4 /                                         |
  || ||1 + cot |-|                                     |                                           |   || |    ||       2/a\                                  ||                                                     |
  \\ \\        \2/                                     /                                           /   || |    ||1 + cot |-|                                  ||                                                     |
                                                                                                       \\ \    \\        \2/                                  //                                                     /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

                                                          //                       /           /pi    \           \\
                                                          ||      0         for And|im(a) = 0, |-- - a| mod pi = 0||
                                                          ||                       \           \2     /           /|
  //   1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                                                       |
- |<                                                  | + |<-(1 + sin(a))                                          |
  \\-cos(a)                  otherwise                /   ||--------------                 otherwise               |
                                                          ||    /a   pi\                                           |
                                                          || tan|- + --|                                           |
                                                          \\    \2   4 /                                           /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\sin{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$

                                                         //                                        /           /pi    \           \\
                                                         ||               0                 for And|im(a) = 0, |-- - a| mod pi = 0||
                                                         ||                                        \           \2     /           /|
                                                         ||                                                                        |
  //  1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   || /         1     \    /a   pi\                                          |
  ||                                                 |   ||-|1 + -----------|*sec|- - --|                                          |
- |< -1                                              | + |< |       /    pi\|    \2   4 /                                          |
  ||------                  otherwise                |   || |    sec|a - --||                                                      |
  \\sec(a)                                           /   || \       \    2 //                                                      |
                                                         ||-------------------------------                 otherwise               |
                                                         ||             /a   pi\                                                   |
                                                         ||          sec|- + --|                                                   |
                                                         \\             \2   4 /                                                   /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$

                                                          //                                    /           /pi    \           \\
                                                          ||             0               for And|im(a) = 0, |-- - a| mod pi = 0||
                                                          ||                                    \           \2     /           /|
                                                          ||                                                                    |
  //   1     for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   || /       /a\\                                                       |
- |<                                                  | + |<-|1 - tan|-||*(1 + sin(a))                                          |
  \\-cos(a)                  otherwise                /   || \       \2//                                                       |
                                                          ||---------------------------                 otherwise               |
                                                          ||                /a\                                                 |
                                                          ||         1 + tan|-|                                                 |
                                                          \\                \2/                                                 /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \frac{\left(1 - \tan{\left(\frac{a}{2} \right)}\right) \left(\sin{\left(a \right)} + 1\right)}{\tan{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$

                                                                                                 //                                                                       /           /pi    \           \\
  //                      1                         for And(im(a) = 0, (pi + a) mod 2*pi = 0)\   ||                              0                                 for And|im(a) = 0, |-- - a| mod pi = 0||
  ||                                                                                         |   ||                                                                       \           \2     /           /|
- |< //  1     for And(im(a) = 0, a mod 2*pi = 0)\                                           | + |<                                                                                                       |
  ||-|<                                          |                  otherwise                |   || /    //  0     for And(im(a) = 0, a mod pi = 0)\\    /a   pi\                                         |
  \\ \\cos(a)              otherwise             /                                           /   ||-|1 + |<                                        ||*cot|- + --|                 otherwise               |
                                                                                                 \\ \    \\sin(a)             otherwise            //    \2   4 /                                         /

$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(- a + \frac{\pi}{2}\right) \bmod \pi = 0 \\- \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(a + \pi\right) \bmod 2 \pi = 0 \\- \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$

  1           1     
------ + -----------
sec(a)   sec(pi - a)

$$\frac{1}{\sec{\left(\pi - a \right)}} + \frac{1}{\sec{\left(a \right)}}$$

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0

$$0$$