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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   / 3*pi\
cos|-----|
   \2 + a/
----------
  sin(a)
cos⁡(3πa+2)sin⁡(a)\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}} 
cos((3*pi)/(2 + a))/sin(a)
Potencias [src] 
    / -3*pi*I    3*pi*I\
    | -------    ------|
    |  2 + a     2 + a |
    |e          e      |
2*I*|-------- + -------|
    \   2          2   /
------------------------
        -I*a    I*a     
     - e     + e
2i(e3iπa+22+e−3iπa+22)eia−e−ia\frac{2 i \left(\frac{e^{\frac{3 i \pi}{a + 2}}}{2} + \frac{e^{- \frac{3 i \pi}{a + 2}}}{2}\right)}{e^{i a} - e^{- i a}} 
2*i*(exp(-3*pi*i/(2 + a))/2 + exp(3*pi*i/(2 + a))/2)/(-exp(-i*a) + exp(i*a))
Parte trigonométrica [src] 
   /    pi\
sec|a - --|
   \    2 /
-----------
    / 3*pi\
 sec|-----|
    \2 + a/
sec⁡(a−π2)sec⁡(3πa+2)\frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}} 
  csc(a)  
----------
   / 3*pi\
sec|-----|
   \2 + a/
csc⁡(a)sec⁡(3πa+2)\frac{\csc{\left(a \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}} 
    / 3*pi\
 cos|-----|
    \2 + a/
-----------
   /    pi\
cos|a - --|
   \    2 /
cos⁡(3πa+2)cos⁡(a−π2)\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} 
/       2/a\\ /        2/   3*pi  \\
|1 + cot |-||*|-1 + cot |---------||
\        \2// \         \2*(2 + a)//
------------------------------------
     /       2/   3*pi  \\    /a\   
   2*|1 + cot |---------||*cot|-|   
     \        \2*(2 + a)//    \2/
(cot⁡2(a2)+1)(cot⁡2(3π2(a+2))−1)2(cot⁡2(3π2(a+2))+1)cot⁡(a2)\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} 
   /pi    \
sec|-- - a|
   \2     /
-----------
    / 3*pi\
 sec|-----|
    \2 + a/
sec⁡(−a+π2)sec⁡(3πa+2)\frac{\sec{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}} 
     csc(a)    
---------------
   /pi    3*pi\
csc|-- - -----|
   \2    2 + a/
csc⁡(a)csc⁡(π2−3πa+2)\frac{\csc{\left(a \right)}}{\csc{\left(\frac{\pi}{2} - \frac{3 \pi}{a + 2} \right)}} 
   /pi    3*pi\
sin|-- + -----|
   \2    2 + a/
---------------
     sin(a)
sin⁡(π2+3πa+2)sin⁡(a)\frac{\sin{\left(\frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}} 
/       2/a\\ /       2/   3*pi  \\
|1 + tan |-||*|1 - tan |---------||
\        \2// \        \2*(2 + a)//
-----------------------------------
     /       2/   3*pi  \\    /a\  
   2*|1 + tan |---------||*tan|-|  
     \        \2*(2 + a)//    \2/
(1−tan⁡2(3π2(a+2)))(tan⁡2(a2)+1)2(tan⁡2(3π2(a+2))+1)tan⁡(a2)\frac{\left(1 - \tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{2 \left(\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}} 
   / 3*pi\       
cos|-----|*csc(a)
   \2 + a/
cos⁡(3πa+2)csc⁡(a)\cos{\left(\frac{3 \pi}{a + 2} \right)} \csc{\left(a \right)} 
cos(3*pi/(2 + a))*csc(a)
Respuesta numérica [src] 
cos((3*pi)/(2 + a))/sin(a)
cos((3*pi)/(2 + a))/sin(a)
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