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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(-x)*cos(pi + x)
-------------------
           /pi    \
sin(-x)*sin|-- + x|
           \2     /
$$\frac{\cos{\left(- x \right)} \cos{\left(x + \pi \right)}}{\sin{\left(- x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}$$ 
(cos(-x)*cos(pi + x))/((sin(-x)*sin(pi/2 + x)))
Simplificación general [src] 
  1   
------
tan(x)
$$\frac{1}{\tan{\left(x \right)}}$$ 
1/tan(x)
Respuesta numérica [src] 
cos(-x)*cos(pi + x)/(sin(-x)*sin(pi/2 + x))
cos(-x)*cos(pi + x)/(sin(-x)*sin(pi/2 + x))
Denominador racional [src] 
        2         
     cos (x)      
------------------
          /    pi\
sin(x)*sin|x + --|
          \    2 /
$$\frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}$$ 
cos(x)^2/(sin(x)*sin(x + pi/2))
Potencias [src] 
cos(x)
------
sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$ 
    / I*x    -I*x\ / I*(pi + x)    I*(-pi - x)\
    |e      e    | |e             e           |
 -4*|---- + -----|*|----------- + ------------|
    \ 2       2  / \     2             2      /
-----------------------------------------------
                 /     /     pi\      /    pi\\
                 |   I*|-x - --|    I*|x + --||
/   I*x    -I*x\ |     \     2 /      \    2 /|
\- e    + e    /*\- e            + e          /
$$- \frac{4 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(\frac{e^{i \left(- x - \pi\right)}}{2} + \frac{e^{i \left(x + \pi\right)}}{2}\right)}{\left(- e^{i x} + e^{- i x}\right) \left(- e^{i \left(- x - \frac{\pi}{2}\right)} + e^{i \left(x + \frac{\pi}{2}\right)}\right)}$$ 
-4*(exp(i*x)/2 + exp(-i*x)/2)*(exp(i*(pi + x))/2 + exp(i*(-pi - x))/2)/((-exp(i*x) + exp(-i*x))*(-exp(i*(-x - pi/2)) + exp(i*(x + pi/2))))
Abrimos la expresión [src] 
cos(x)
------
sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$ 
cos(x)/sin(x)
Parte trigonométrica [src] 
 /       2/x   pi\\ /       1   \ /       2/x\\ 
-|1 + tan |- + --||*|1 - -------|*|1 - tan |-|| 
 \        \2   4 // |       2/x\| \        \2// 
                    |    tan |-||               
                    \        \2//               
------------------------------------------------
         /       1   \    /x\    /x   pi\       
       4*|1 + -------|*tan|-|*tan|- + --|       
         |       2/x\|    \2/    \2   4 /       
         |    tan |-||                          
         \        \2//
$$- \frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{4 \left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
                3                
    /       /x\\  /      1   \   
   -|1 + tan|-|| *|1 - ------|   
    \       \2//  |       /x\|   
                  |    tan|-||   
                  \       \2//   
---------------------------------
                /           /x\ \
                |      2*tan|-| |
  /       2/x\\ |           \2/ |
2*|1 + tan |-||*|1 + -----------|
  \        \2// |           2/x\|
                |    1 + tan |-||
                \            \2//
$$- \frac{\left(1 - \frac{1}{\tan{\left(\frac{x}{2} \right)}}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{3}}{2 \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
             3                              
 /       /x\\     2/x\    2/x\ /       /x\\ 
-|1 + tan|-|| *cot |-|*sin |-|*|1 - cot|-|| 
 \       \2//      \2/     \2/ \       \2// 
--------------------------------------------
               2*(1 + sin(x))
$$- \frac{\left(1 - \cot{\left(\frac{x}{2} \right)}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{3} \sin^{2}{\left(\frac{x}{2} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{2 \left(\sin{\left(x \right)} + 1\right)}$$ 
                  3                                      
 /       /x   pi\\  /          /x\  \                    
 |    cos|- - --||  |       cos|-|  |                    
 |       \2   2 /|  |          \2/  |    /x\    /    pi\ 
-|1 + -----------| *|1 - -----------|*cos|-|*cos|x - --| 
 |          /x\  |  |       /x   pi\|    \2/    \    2 / 
 |       cos|-|  |  |    cos|- - --||                    
 \          \2/  /  \       \2   2 //                    
---------------------------------------------------------
               /       /    pi\\    /x   pi\             
             4*|1 + cos|x - --||*cos|- - --|             
               \       \    2 //    \2   2 /
$$- \frac{\left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{3} \left(- \frac{\cos{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{4 \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$ 
                    2/x\
                 sin |-|
    sin(x)           \2/
-------------- - -------
2*(1 - cos(x))    sin(x)
$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + \frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)}$$ 
cot(x)
$$\cot{\left(x \right)}$$ 
       /x\         /pi   x\
    csc|-|      csc|-- - -|
       \2/         \2    2/
------------- - -----------
     /pi   x\          /x\ 
2*csc|-- - -|     2*csc|-| 
     \2    2/          \2/
$$\frac{\csc{\left(\frac{x}{2} \right)}}{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{x}{2} \right)}}$$ 
                3                         
 /         2/x\\                          
 |    2*sin |-||                          
 |          \2/|     2    /      sin(x) \ 
-|1 + ---------| *sin (x)*|1 - ---------| 
 \      sin(x) /          |         2/x\| 
                          |    2*sin |-|| 
                          \          \2// 
------------------------------------------
                            2/x\          
          8*(1 + sin(x))*sin |-|          
                             \2/
$$- \frac{\left(1 - \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{3} \sin^{2}{\left(x \right)}}{8 \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}$$ 
cos(x)
------
sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$ 
 sin(2*x)
---------
     2   
2*sin (x)
$$\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}$$ 
       /x\         /x   pi\
    cos|-|      cos|- - --|
       \2/         \2   2 /
------------- - -----------
     /x   pi\          /x\ 
2*cos|- - --|     2*cos|-| 
     \2   2 /          \2/
$$\frac{\cos{\left(\frac{x}{2} \right)}}{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{x}{2} \right)}}$$ 
             3                      
 /      1   \     2/x\ /       /x\\ 
-|1 + ------| *cot |-|*|1 - cot|-|| 
 |       /x\|      \2/ \       \2// 
 |    cot|-||                       
 \       \2//                       
------------------------------------
                 /           /x\ \  
                 |      2*cot|-| |  
   /       2/x\\ |           \2/ |  
 2*|1 + cot |-||*|1 + -----------|  
   \        \2// |           2/x\|  
                 |    1 + cot |-||  
                 \            \2//
$$- \frac{\left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{3} \left(1 - \cot{\left(\frac{x}{2} \right)}\right) \cot^{2}{\left(\frac{x}{2} \right)}}{2 \left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
                  3                               
 /          /x\  \  /       /x   pi\\             
 |       sec|-|  |  |    sec|- - --||             
 |          \2/  |  |       \2   2 /|    /x   pi\ 
-|1 + -----------| *|1 - -----------|*sec|- - --| 
 |       /x   pi\|  |          /x\  |    \2   2 / 
 |    sec|- - --||  |       sec|-|  |             
 \       \2   2 //  \          \2/  /             
--------------------------------------------------
        /         1     \    /x\    /    pi\      
      4*|1 + -----------|*sec|-|*sec|x - --|      
        |       /    pi\|    \2/    \    2 /      
        |    sec|x - --||                         
        \       \    2 //
$$- \frac{\left(1 - \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{x}{2} \right)}}\right) \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{4 \left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$ 
   cos(x)  
-----------
   /    pi\
cos|x - --|
   \    2 /
$$\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}$$ 
    /    3*pi\    /pi    \ 
-sin|x + ----|*sin|-- - x| 
    \     2  /    \2     / 
---------------------------
       cos(x)*sin(x)
$$- \frac{\sin{\left(- x + \frac{\pi}{2} \right)} \sin{\left(x + \frac{3 \pi}{2} \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$ 
 /       2/x   pi\\ /        1   \ /        2/x\\ 
-|1 + cot |- + --||*|-1 + -------|*|-1 + cot |-|| 
 \        \2   4 // |        2/x\| \         \2// 
                    |     cot |-||                
                    \         \2//                
--------------------------------------------------
          /       1   \    /x\    /x   pi\        
        4*|1 + -------|*cot|-|*cot|- + --|        
          |       2/x\|    \2/    \2   4 /        
          |    cot |-||                           
          \        \2//
$$- \frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{4 \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
csc(x)
------
sec(x)
$$\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}}$$ 
   /x\      /x\
cot|-|   tan|-|
   \2/      \2/
------ - ------
  2        2
$$- \frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{\cot{\left(\frac{x}{2} \right)}}{2}$$ 
        2         
     cos (x)      
------------------
          /    pi\
sin(x)*sin|x + --|
          \    2 /
$$\frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}$$ 
              /x\
           tan|-|
   1          \2/
-------- - ------
     /x\     2   
2*tan|-|         
     \2/
$$- \frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{x}{2} \right)}}$$ 
  1   
------
tan(x)
$$\frac{1}{\tan{\left(x \right)}}$$ 
   /    pi\
sec|x - --|
   \    2 /
-----------
   sec(x)
$$\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}$$ 
             3                      
 /       /x\\     2/x\ /      1   \ 
-|1 + tan|-|| *cos |-|*|1 - ------| 
 \       \2//      \2/ |       /x\| 
                       |    tan|-|| 
                       \       \2// 
------------------------------------
           2*(1 + sin(x))
$$- \frac{\left(1 - \frac{1}{\tan{\left(\frac{x}{2} \right)}}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{3} \cos^{2}{\left(\frac{x}{2} \right)}}{2 \left(\sin{\left(x \right)} + 1\right)}$$ 
             3                     
 /       /x\\  /      1   \        
-|1 + tan|-|| *|1 - ------|*sin(x) 
 \       \2//  |       /x\|        
               |    tan|-||        
               \       \2//        
-----------------------------------
                         /x\       
       4*(1 + sin(x))*tan|-|       
                         \2/
$$- \frac{\left(1 - \frac{1}{\tan{\left(\frac{x}{2} \right)}}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{3} \sin{\left(x \right)}}{4 \left(\sin{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}$$ 
    /     pi\ 
-sec|-x - --| 
    \     2 / 
--------------
    sec(x)
$$- \frac{\sec{\left(- x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}$$ 
     2/x\            
  sin |-|            
      \2/     sin(x) 
- ------- + ---------
   sin(x)        2/x\
            4*sin |-|
                  \2/
$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + \frac{\sin{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}}$$ 
   /x\           
cot|-|           
   \2/      1    
------ - --------
  2           /x\
         2*cot|-|
              \2/
$$\frac{\cot{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2 \cot{\left(\frac{x}{2} \right)}}$$ 
 /       2/x   pi\\ /       2/x\\ /       2/x\\ 
-|1 + tan |- + --||*|1 - cot |-||*|1 - tan |-|| 
 \        \2   4 // \        \2// \        \2// 
------------------------------------------------
         /       2/x\\    /x\    /x   pi\       
       4*|1 + cot |-||*tan|-|*tan|- + --|       
         \        \2//    \2/    \2   4 /
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
          /pi    \
csc(x)*csc|-- - x|
          \2     /
------------------
        2         
     sec (x)
$$\frac{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}$$ 
   /x   pi\          /x\   
sec|- - --|       sec|-|   
   \2   2 /          \2/   
----------- - -------------
       /x\         /x   pi\
  2*sec|-|    2*sec|- - --|
       \2/         \2   2 /
$$- \frac{\sec{\left(\frac{x}{2} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{x}{2} \right)}}$$ 
                                      3                          
 /                  ___    /      pi\\  /       ___    /    pi\\ 
-|-1 - 2*sin(x) + \/ 2 *sin|2*x + --|| *|-1 + \/ 2 *sin|x + --|| 
 \                         \      4 //  \              \    4 // 
-----------------------------------------------------------------
                          2                           2          
          8*(1 - cos(2*x)) *(1 + sin(x))*(-1 + cos(x))
$$- \frac{\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 1\right) \left(- 2 \sin{\left(x \right)} + \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} - 1\right)^{3}}{8 \left(1 - \cos{\left(2 x \right)}\right)^{2} \left(\sin{\left(x \right)} + 1\right) \left(\cos{\left(x \right)} - 1\right)^{2}}$$ 
  -cos(x)   
------------
   /     pi\
cos|-x - --|
   \     2 /
$$- \frac{\cos{\left(x \right)}}{\cos{\left(- x - \frac{\pi}{2} \right)}}$$ 
                  3                          
 /       /pi   x\\  /          /x\  \        
 |    csc|-- - -||  |       csc|-|  |        
 |       \2    2/|  |          \2/  |    /x\ 
-|1 + -----------| *|1 - -----------|*csc|-| 
 |          /x\  |  |       /pi   x\|    \2/ 
 |       csc|-|  |  |    csc|-- - -||        
 \          \2/  /  \       \2    2//        
---------------------------------------------
        /      1   \           /pi   x\      
      4*|1 + ------|*csc(x)*csc|-- - -|      
        \    csc(x)/           \2    2/
$$- \frac{\left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{3} \left(- \frac{\csc{\left(\frac{x}{2} \right)}}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(\frac{x}{2} \right)}}{4 \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$ 
csc(x)*sec(x)
-------------
    2/pi    \
 csc |-- - x|
     \2     /
$$\frac{\csc{\left(x \right)} \sec{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$ 
   csc(x)  
-----------
   /pi    \
csc|-- - x|
   \2     /
$$\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
 /       2/x   pi\\ /        2/x\\ /        2/x\\ 
-|1 + cot |- + --||*|-1 + cot |-||*|-1 + tan |-|| 
 \        \2   4 // \         \2// \         \2// 
--------------------------------------------------
          /       2/x\\    /x\    /x   pi\        
        4*|1 + tan |-||*cot|-|*cot|- + --|        
          \        \2//    \2/    \2   4 /
$$- \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{4 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$ 
-(1 + cot(x/2 + pi/4)^2)*(-1 + cot(x/2)^2)*(-1 + tan(x/2)^2)/(4*(1 + tan(x/2)^2)*cot(x/2)*cot(x/2 + pi/4))
Unión de expresiones racionales [src] 
         2          
      cos (x)       
--------------------
          /pi + 2*x\
sin(x)*sin|--------|
          \   2    /
$$\frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \sin{\left(\frac{2 x + \pi}{2} \right)}}$$ 
cos(x)^2/(sin(x)*sin((pi + 2*x)/2))
Combinatoria [src] 
cos(x)
------
sin(x)
$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$ 
cos(x)/sin(x)
Denominador común [src] 
cos(-x)*cos(pi + x)
-------------------
   cos(x)*sin(-x)
$$\frac{\cos{\left(- x \right)} \cos{\left(x + \pi \right)}}{\sin{\left(- x \right)} \cos{\left(x \right)}}$$ 
cos(-x)*cos(pi + x)/(cos(x)*sin(-x))
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