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Simplificación de expresiones/ Descomposición en fracciones simples/ cos((\pi)/(2)+a)cos(\pi-a)+sin((3\pi)/(2)+a)sin(\pi+a)

¿Cómo vas a descomponer esta cos((\pi)/(2)+a)cos(\pi-a)+sin((3\pi)/(2)+a)sin(\pi+a) expresión en fracciones?

Solución

Ha introducido [src]

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

\sin{\left(a + \pi \right)} \sin{\left(a + \frac{3 \frac{1}{\pi}}{2} \right)} + \cos{\left(\pi - a \right)} \cos{\left(a + \frac{\pi}{2} \right)}

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

Simplificación general [src]

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

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

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

Respuesta numérica [src]

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

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

Combinatoria [src]

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

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

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

Potencias [src]

                            /   /    pi\      /     pi\\                                  /     /      3  \      /     3  \\
                            | I*|a + --|    I*|-a - --||                                  |   I*|-a - ----|    I*|a + ----||
/ I*(pi - a)    I*(a - pi)\ |   \    2 /      \     2 /|   /   I*(-pi - a)    I*(pi + a)\ |     \     2*pi/      \    2*pi/|
|e             e          | |e             e           |   \- e            + e          /*\- e              + e            /
|----------- + -----------|*|----------- + ------------| - -----------------------------------------------------------------
\     2             2     / \     2             2      /                                   4

\left(\frac{e^{i \left(\pi - a\right)}}{2} + \frac{e^{i \left(a - \pi\right)}}{2}\right) \left(\frac{e^{i \left(- a - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(a + \frac{\pi}{2}\right)}}{2}\right) - \frac{\left(- e^{i \left(- a - \frac{3}{2 \pi}\right)} + e^{i \left(a + \frac{3}{2 \pi}\right)}\right) \left(- e^{i \left(- a - \pi\right)} + e^{i \left(a + \pi\right)}\right)}{4}

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

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

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

Unión de expresiones racionales [src]

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

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

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

Denominador racional [src]

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

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

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

Abrimos la expresión [src]

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

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

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

Denominador común [src]

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

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

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

Parte trigonométrica [src]

   /        2   \                                                                   
 2*|-1 + -------|                                                                   
   |        2/a\|                                 /a    3  \                        
   |     sec |-||                            4*sec|- + ----|                        
   \         \2//                                 \2   4*pi/                        
------------------ - ---------------------------------------------------------------
   /a\    /a   pi\   /          2/a    3  \  \                                      
sec|-|*sec|- - --|   |       sec |- + ----|  |                                      
   \2/    \2   2 /   |           \2   4*pi/  |    /a\    /a   pi\    /a   pi    3  \
                     |1 + -------------------|*sec|-|*sec|- - --|*sec|- - -- + ----|
                     |       2/a   pi    3  \|    \2/    \2   2 /    \2   2    4*pi/
                     |    sec |- - -- + ----||                                      
                     \        \2   2    4*pi//

\frac{2 \left(-1 + \frac{2}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)}{\sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{4 \sec{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}} + 1\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}

                                                    /          2/a\   \ /        2/a   pi\\
                                                    |       sec |-|   | |     sec |- - --||
                                  /         1     \ |           \2/   | |         \2   4 /|
                                  |1 + -----------|*|-1 + ------------|*|-1 + ------------|
                                  |       /    pi\| |        2/a   pi\| |        2/a   pi\|
                                  |    sec|a - --|| |     sec |- - --|| |     sec |- + --||
        1                1        \       \    2 // \         \2   2 // \         \2   4 //
----------------- - ----------- + ---------------------------------------------------------
     /       3  \        / 3  \                                2/a\                        
2*sec|2*a + ----|   2*sec|----|                           2*sec |-|                        
     \      2*pi/        \2*pi/                                 \2/

\frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - 1\right)}{2 \sec^{2}{\left(\frac{a}{2} \right)}} - \frac{1}{2 \sec{\left(\frac{3}{2 \pi} \right)}} + \frac{1}{2 \sec{\left(2 a + \frac{3}{2 \pi} \right)}}

                /a\    /a    3  \    /a\    /a    3  \
           4*cos|-|*sec|- + ----|*sin|-|*sin|- + ----|
sin(2*a)        \2/    \2   4*pi/    \2/    \2   4*pi/
-------- - -------------------------------------------
   2                           2/a    3  \            
                        1 + tan |- + ----|            
                                \2   4*pi/

\frac{\sin{\left(2 a \right)}}{2} - \frac{4 \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cos{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}

                 2/a\    4/a\    /a\    /a    3  \
           16*cot |-|*sin |-|*cot|-|*tan|- + ----|
sin(2*a)          \4/     \4/    \2/    \2   4*pi/
-------- - ---------------------------------------
   2                         2/a    3  \          
                      1 + tan |- + ----|          
                              \2   4*pi/

\frac{\sin{\left(2 a \right)}}{2} - \frac{16 \sin^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}

/       1   \ /       2/a   pi\\                                            
|1 - -------|*|1 - tan |- + --||                                            
|       2/a\| \        \2   4 //                     /a    3  \             
|    tan |-||                                   4*tan|- + ----|             
\        \2//                                        \2   4*pi/             
-------------------------------- - -----------------------------------------
/       1   \ /       2/a   pi\\   /       1   \ /       2/a    3  \\    /a\
|1 + -------|*|1 + tan |- + --||   |1 + -------|*|1 + tan |- + ----||*tan|-|
|       2/a\| \        \2   4 //   |       2/a\| \        \2   4*pi//    \2/
|    tan |-||                      |    tan |-||                            
\        \2//                      \        \2//

\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}

        1                     1          
------------------ - --------------------
          /pi    \             /     3  \
sec(a)*sec|-- - a|   csc(a)*csc|a + ----|
          \2     /             \    2*pi/

\frac{1}{\sec{\left(a \right)} \sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}}

      1                  1          
------------- - --------------------
csc(a)*sec(a)             /     3  \
                csc(a)*csc|a + ----|
                          \    2*pi/

\frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}}

       /       3  \      / 3  \      2/a\              /        2/a   pi\\ /        2/a\\
  - cos|2*a + ----| + cos|----|   cos |-|*(1 + sin(a))*|-1 + cot |- + --||*|-1 + tan |-||
       \      2*pi/      \2*pi/       \2/              \         \2   4 // \         \2//
- ----------------------------- + -------------------------------------------------------
                2                                            2

\frac{\left(\sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{2} - \frac{- \cos{\left(2 a + \frac{3}{2 \pi} \right)} + \cos{\left(\frac{3}{2 \pi} \right)}}{2}

  /          2      \                                                             
2*|-1 + ------------|                                                             
  |        2/pi   a\|                           /pi    3     a\                   
  |     csc |-- - -||                      4*csc|-- - ---- - -|                   
  \         \2    2//                           \2    4*pi   2/                   
--------------------- - ----------------------------------------------------------
     /a\    /pi   a\    /       2/pi    3     a\\                                 
  csc|-|*csc|-- - -|    |    csc |-- - ---- - -||                                 
     \2/    \2    2/    |        \2    4*pi   2/|    /a\    /pi   a\    /a    3  \
                        |1 + -------------------|*csc|-|*csc|-- - -|*csc|- + ----|
                        |          2/a    3  \  |    \2/    \2    2/    \2   4*pi/
                        |       csc |- + ----|  |                                 
                        \           \2   4*pi/  /

\frac{2 \left(-1 + \frac{2}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}\right)}{\csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{4 \csc{\left(- \frac{a}{2} - \frac{3}{4 \pi} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{3}{4 \pi} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)} \csc{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}

                                             /a\    /a   pi\    /a   pi    3  \
                                        4*cos|-|*cos|- - --|*cos|- - -- + ----|
  /          2/a\\    /a\    /a   pi\        \2/    \2   2 /    \2   2    4*pi/
2*|-1 + 2*cos |-||*cos|-|*cos|- - --| - ---------------------------------------
  \           \2//    \2/    \2   2 /   /       2/a   pi    3  \\              
                                        |    cos |- - -- + ----||              
                                        |        \2   2    4*pi/|    /a    3  \
                                        |1 + -------------------|*cos|- + ----|
                                        |          2/a    3  \  |    \2   4*pi/
                                        |       cos |- + ----|  |              
                                        \           \2   4*pi/  /

2 \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - \frac{4 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}

  1              1         
------ - ------------------
sec(a)      /    pi    3  \
         sec|a - -- + ----|
            \    2    2*pi/
---------------------------
           /    pi\        
        sec|a - --|        
           \    2 /

\frac{- \frac{1}{\sec{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}} + \frac{1}{\sec{\left(a \right)}}}{\sec{\left(a - \frac{\pi}{2} \right)}}

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

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

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

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

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

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

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

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

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

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

/        1   \ /        2/a   pi\\                                            
|-1 + -------|*|-1 + cot |- + --||                                            
|        2/a\| \         \2   4 //                     /a    3  \             
|     cot |-||                                    4*cot|- + ----|             
\         \2//                                         \2   4*pi/             
---------------------------------- - -----------------------------------------
 /       1   \ /       2/a   pi\\    /       1   \ /       2/a    3  \\    /a\
 |1 + -------|*|1 + cot |- + --||    |1 + -------|*|1 + cot |- + ----||*cot|-|
 |       2/a\| \        \2   4 //    |       2/a\| \        \2   4*pi//    \2/
 |    cot |-||                       |    cot |-||                            
 \        \2//                       \        \2//

\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}

  /       2/a\         /a    3  \  \       
  |1 - tan |-|    2*tan|- + ----|  |       
  |        \2/         \2   4*pi/  |    /a\
2*|----------- - ------------------|*tan|-|
  |       2/a\          2/a    3  \|    \2/
  |1 + tan |-|   1 + tan |- + ----||       
  \        \2/           \2   4*pi//       
-------------------------------------------
                       2/a\                
                1 + tan |-|                
                        \2/

\frac{2 \left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}

        1                     1          
------------------ - --------------------
          /pi    \             /     3  \
csc(a)*csc|-- - a|   csc(a)*csc|a + ----|
          \2     /             \    2*pi/

- \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}

/        2/a   pi\\ /        2/a\\              /a    3  \    /a\      
|-1 + cot |- + --||*|-1 + tan |-||         4*cot|- + ----|*tan|-|      
\         \2   4 // \         \2//              \2   4*pi/    \2/      
---------------------------------- - ----------------------------------
 /       2/a   pi\\ /       2/a\\    /       2/a    3  \\ /       2/a\\
 |1 + cot |- + --||*|1 + tan |-||    |1 + cot |- + ----||*|1 + tan |-||
 \        \2   4 // \        \2//    \        \2   4*pi// \        \2//

\frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right)}

                      2/a    3  \              
                 4*sin |- + ----|*sin(a)       
sin(2*a)               \2   4*pi/              
-------- - ------------------------------------
   2       /         4/a    3  \\              
           |    4*sin |- + ----||              
           |          \2   4*pi/|    /     3  \
           |1 + ----------------|*sin|a + ----|
           |        2/     3  \ |    \    2*pi/
           |     sin |a + ----| |              
           \         \    2*pi/ /

\frac{\sin{\left(2 a \right)}}{2} - \frac{4 \sin{\left(a \right)} \sin^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sin^{2}{\left(a + \frac{3}{2 \pi} \right)}} + 1\right) \sin{\left(a + \frac{3}{2 \pi} \right)}}

/       2/a\\ /       2/a   pi\\              /a\    /a    3  \      
|1 - cot |-||*|1 - tan |- + --||         4*cot|-|*tan|- + ----|      
\        \2// \        \2   4 //              \2/    \2   4*pi/      
-------------------------------- - ----------------------------------
/       2/a\\ /       2/a   pi\\   /       2/a\\ /       2/a    3  \\
|1 + cot |-||*|1 + tan |- + --||   |1 + cot |-||*|1 + tan |- + ----||
\        \2// \        \2   4 //   \        \2// \        \2   4*pi//

\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right) \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}

             /       /     3  \\    /a\    /a\    /a    3  \
           2*|1 + cos|a + ----||*cos|-|*sin|-|*sin|- + ----|
sin(2*a)     \       \    2*pi//    \2/    \2/    \2   4*pi/
-------- - -------------------------------------------------
   2                            /a    3  \                  
                             cos|- + ----|                  
                                \2   4*pi/

- \frac{2 \left(\cos{\left(a + \frac{3}{2 \pi} \right)} + 1\right) \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cos{\left(\frac{a}{2} \right)}}{\cos{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}} + \frac{\sin{\left(2 a \right)}}{2}

                                        2/a\    /a\    /a    3  \
                                   4*sin |-|*cot|-|*tan|- + ----|
  /          2/a\\    /a\    /a\         \2/    \2/    \2   4*pi/
2*|-1 + 2*cos |-||*cos|-|*sin|-| - ------------------------------
  \           \2//    \2/    \2/                2/a    3  \      
                                         1 + tan |- + ----|      
                                                 \2   4*pi/

2 \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)} - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}

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

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

  /        2/a\         /a    3  \  \       
  |-1 + cot |-|    2*cot|- + ----|  |       
  |         \2/         \2   4*pi/  |    /a\
2*|------------ - ------------------|*cot|-|
  |       2/a\           2/a    3  \|    \2/
  |1 + cot |-|    1 + cot |- + ----||       
  \        \2/            \2   4*pi//       
--------------------------------------------
                       2/a\                 
                1 + cot |-|                 
                        \2/

\frac{2 \left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}\right) \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}

                                                         /        2/pi   a\\ /         2/a   pi\ \
                                                         |     csc |-- - -|| |      csc |- + --| |
                                            /      1   \ |         \2    2/| |          \2   4 / |
                                            |1 + ------|*|-1 + ------------|*|-1 + --------------|
                                            \    csc(a)/ |          2/a\   | |        2/  a   pi\|
                                                         |       csc |-|   | |     csc |- - + --||
          1                     1                        \           \2/   / \         \  2   4 //
---------------------- - ---------------- + ------------------------------------------------------
     /pi          3  \        /pi    3  \                            2/pi   a\                    
2*csc|-- - 2*a - ----|   2*csc|-- - ----|                       2*csc |-- - -|                    
     \2          2*pi/        \2    2*pi/                             \2    2/

\frac{\left(-1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(-1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(a \right)}}\right)}{2 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(- \frac{3}{2 \pi} + \frac{\pi}{2} \right)}} + \frac{1}{2 \csc{\left(- 2 a - \frac{3}{2 \pi} + \frac{\pi}{2} \right)}}

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

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

                                                             /           /a\ \                                   
                                                           2 |      2*tan|-| |                                   
                                              /       2/a\\  |           \2/ | /        2/a\\ /          1      \
                                              |1 - tan |-|| *|1 + -----------|*|-1 + tan |-||*|-1 + ------------|
         2/     3  \              2/ 3  \     \        \4//  |           2/a\| \         \2// |        2/a   pi\|
  1 - tan |a + ----|       1 - tan |----|                    |    1 + tan |-||                |     tan |- + --||
          \    4*pi/               \4*pi/                    \            \2//                \         \2   4 //
---------------------- - ------------------ + -------------------------------------------------------------------
  /       2/     3  \\     /       2/ 3  \\                                            2                         
2*|1 + tan |a + ----||   2*|1 + tan |----||                               /       2/a\\                          
  \        \    4*pi//     \        \4*pi//                             2*|1 + tan |-||                          
                                                                          \        \4//

\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right)}{2 \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{1 - \tan^{2}{\left(a + \frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(a + \frac{3}{4 \pi} \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(\frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(\frac{3}{4 \pi} \right)} + 1\right)}

   /       3  \      / 3  \        2/a\              
cos|2*a + ----|   cos|----|   2*cos |-|*cos(a)*sin(a)
   \      2*pi/      \2*pi/         \2/              
--------------- - --------- + -----------------------
       2              2              1 + cos(a)

\frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2} + \frac{2 \sin{\left(a \right)} \cos^{2}{\left(\frac{a}{2} \right)} \cos{\left(a \right)}}{\cos{\left(a \right)} + 1}

                 4/a\    2/a\    /a    3  \
           16*cos |-|*tan |-|*tan|- + ----|
sin(2*a)          \4/     \4/    \2   4*pi/
-------- - --------------------------------
   2         /       2/a    3  \\    /a\   
             |1 + tan |- + ----||*tan|-|   
             \        \2   4*pi//    \2/

\frac{\sin{\left(2 a \right)}}{2} - \frac{16 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}

                                                                  /          4/a\\                      
                                                                  |     4*sin |-|| /           2       \
                                           2/pi   a\              |           \2/| |        cos (a)    |
                                        sin |-- + -|*(1 + sin(a))*|-1 + ---------|*|-1 + --------------|
   /pi          3  \      /pi    3  \       \2    2/              |         2    | |          4/a   pi\|
sin|-- + 2*a + ----|   sin|-- + ----|                             \      sin (a) / |     4*sin |- + --||
   \2          2*pi/      \2    2*pi/                                              \           \2   4 //
-------------------- - -------------- + ----------------------------------------------------------------
         2                   2                                         2

\frac{\left(-1 + \frac{\cos^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} - 1\right) \left(\sin{\left(a \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\sin{\left(2 a + \frac{3}{2 \pi} + \frac{\pi}{2} \right)}}{2} - \frac{\sin{\left(\frac{3}{2 \pi} + \frac{\pi}{2} \right)}}{2}

                                                        2/a    3  \              
                                                   4*sin |- + ----|*sin(a)       
  /          2/pi   a\\    /a\    /pi   a\               \2   4*pi/              
2*|-1 + 2*sin |-- + -||*sin|-|*sin|-- + -| - ------------------------------------
  \           \2    2//    \2/    \2    2/   /         4/a    3  \\              
                                             |    4*sin |- + ----||              
                                             |          \2   4*pi/|    /     3  \
                                             |1 + ----------------|*sin|a + ----|
                                             |        2/     3  \ |    \    2*pi/
                                             |     sin |a + ----| |              
                                             \         \    2*pi/ /

2 \left(2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - 1\right) \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - \frac{4 \sin{\left(a \right)} \sin^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sin^{2}{\left(a + \frac{3}{2 \pi} \right)}} + 1\right) \sin{\left(a + \frac{3}{2 \pi} \right)}}

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

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

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

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

                                                              /           /a\ \                                   
                                                            2 |      2*cot|-| |                                   
                                              /        2/a\\  |           \2/ | /        1   \ /        2/a   pi\\
                                              |-1 + cot |-|| *|1 + -----------|*|-1 + -------|*|-1 + cot |- + --||
         2/     3  \              2/ 3  \     \         \4//  |           2/a\| |        2/a\| \         \2   4 //
 -1 + cot |a + ----|      -1 + cot |----|                     |    1 + cot |-|| |     cot |-||                    
          \    4*pi/               \4*pi/                     \            \2// \         \2//                    
---------------------- - ------------------ + --------------------------------------------------------------------
  /       2/     3  \\     /       2/ 3  \\                                            2                          
2*|1 + cot |a + ----||   2*|1 + cot |----||                               /       2/a\\                           
  \        \    4*pi//     \        \4*pi//                             2*|1 + cot |-||                           
                                                                          \        \4//

\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(a + \frac{3}{4 \pi} \right)} - 1}{2 \left(\cot^{2}{\left(a + \frac{3}{4 \pi} \right)} + 1\right)} - \frac{-1 + \cot^{2}{\left(\frac{3}{4 \pi} \right)}}{2 \left(1 + \cot^{2}{\left(\frac{3}{4 \pi} \right)}\right)}

                                            2                                                        
   /       3  \      / 3  \   /        2/a\\     4/a\              /        2/a   pi\\ /        2/a\\
cos|2*a + ----|   cos|----|   |-1 + cot |-|| *sin |-|*(1 + sin(a))*|-1 + cot |- + --||*|-1 + tan |-||
   \      2*pi/      \2*pi/   \         \4//      \4/              \         \2   4 // \         \2//
--------------- - --------- + -----------------------------------------------------------------------
       2              2                                          2

\frac{\left(\sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \sin^{4}{\left(\frac{a}{4} \right)}}{2} + \frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2}

          /        /a    3  \           \       
          |   2*tan|- + ----|           |       
     2/a\ |        \2   4*pi/           |    /a\
2*cos |-|*|- ------------------ + cos(a)|*tan|-|
      \2/ |         2/a    3  \         |    \2/
          |  1 + tan |- + ----|         |       
          \          \2   4*pi/         /

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

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

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

     1              1      
----------- - -------------
   /pi    \      /     3  \
csc|-- - a|   csc|a + ----|
   \2     /      \    2*pi/
---------------------------
           csc(a)

\frac{- \frac{1}{\csc{\left(a + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}}{\csc{\left(a \right)}}

                                                        /        2/a   pi\\ /        2/a   pi\\
                                                        |     cos |- - --|| |     cos |- + --||
                                 2/a\ /       /    pi\\ |         \2   2 /| |         \2   4 /|
                              cos |-|*|1 + cos|a - --||*|-1 + ------------|*|-1 + ------------|
   /       3  \      / 3  \       \2/ \       \    2 // |          2/a\   | |        2/a   pi\|
cos|2*a + ----|   cos|----|                             |       cos |-|   | |     cos |- - --||
   \      2*pi/      \2*pi/                             \           \2/   / \         \2   4 //
--------------- - --------- + -----------------------------------------------------------------
       2              2                                       2

\frac{\left(-1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(-1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}\right) \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{2} + \frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2}

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

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

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

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

      1                       1               
------------- + ------------------------------
csc(a)*sec(a)      /    pi\    /    pi    3  \
                sec|a + --|*sec|a - -- + ----|
                   \    2 /    \    2    2*pi/

\frac{1}{\sec{\left(a + \frac{\pi}{2} \right)} \sec{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}}

1/(csc(a)*sec(a)) + 1/(sec(a + pi/2)*sec(a - pi/2 + 3/(2*pi)))

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