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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x+y)/((sin(x)*sin(y)))

¿Cómo vas a descomponer esta sin(x+y)/((sin(x)*sin(y))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(x + y) 
-------------
sin(x)*sin(y)
sin⁡(x+y)sin⁡(x)sin⁡(y)\frac{\sin{\left(x + y \right)}}{\sin{\left(x \right)} \sin{\left(y \right)}} 
sin(x + y)/((sin(x)*sin(y)))
Simplificación general [src] 
  1        1   
------ + ------
tan(x)   tan(y)
1tan⁡(y)+1tan⁡(x)\frac{1}{\tan{\left(y \right)}} + \frac{1}{\tan{\left(x \right)}} 
1/tan(x) + 1/tan(y)
Respuesta numérica [src] 
sin(x + y)/(sin(x)*sin(y))
sin(x + y)/(sin(x)*sin(y))
Potencias [src] 
     /   I*(-x - y)    I*(x + y)\
 2*I*\- e           + e         /
---------------------------------
/   -I*x    I*x\ /   -I*y    I*y\
\- e     + e   /*\- e     + e   /
2i(−ei(−x−y)+ei(x+y))(eix−e−ix)(eiy−e−iy)\frac{2 i \left(- e^{i \left(- x - y\right)} + e^{i \left(x + y\right)}\right)}{\left(e^{i x} - e^{- i x}\right) \left(e^{i y} - e^{- i y}\right)} 
2*i*(-exp(i*(-x - y)) + exp(i*(x + y)))/((-exp(-i*x) + exp(i*x))*(-exp(-i*y) + exp(i*y)))
Abrimos la expresión [src] 
cos(x)   cos(y)
------ + ------
sin(x)   sin(y)
cos⁡(y)sin⁡(y)+cos⁡(x)sin⁡(x)\frac{\cos{\left(y \right)}}{\sin{\left(y \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} 
cos(x)/sin(x) + cos(y)/sin(y)
Parte trigonométrica [src] 
   /    pi\      /    pi\
sec|x - --|   sec|y - --|
   \    2 /      \    2 /
----------- + -----------
   sec(x)        sec(y)
sec⁡(y−π2)sec⁡(y)+sec⁡(x−π2)sec⁡(x)\frac{\sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} 
cot(x) + cot(y)
cot⁡(x)+cot⁡(y)\cot{\left(x \right)} + \cot{\left(y \right)} 
       /x\             /y\         /x   pi\      /y   pi\
    cos|-|          cos|-|      cos|- - --|   cos|- - --|
       \2/             \2/         \2   2 /      \2   2 /
------------- + ------------- - ----------- - -----------
     /x   pi\        /y   pi\          /x\           /y\ 
2*cos|- - --|   2*cos|- - --|     2*cos|-|      2*cos|-| 
     \2   2 /        \2   2 /          \2/           \2/
cos⁡(x2)2cos⁡(x2−π2)+cos⁡(y2)2cos⁡(y2−π2)−cos⁡(y2−π2)2cos⁡(y2)−cos⁡(x2−π2)2cos⁡(x2)\frac{\cos{\left(\frac{x}{2} \right)}}{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{y}{2} \right)}}{2 \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{y}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{x}{2} \right)}} 
csc(x)*csc(y)
-------------
  csc(x + y)
csc⁡(x)csc⁡(y)csc⁡(x+y)\frac{\csc{\left(x \right)} \csc{\left(y \right)}}{\csc{\left(x + y \right)}} 
                                   2                        2    
                      /       2/x\\            /       2/y\\     
                      |1 + tan |-||            |1 + tan |-||     
  1        1          \        \4//            \        \4//     
------ + ------ - ---------------------- - ----------------------
   /x\      /y\     /       2/x\\    /x\     /       2/y\\    /y\
tan|-|   tan|-|   4*|1 - tan |-||*tan|-|   4*|1 - tan |-||*tan|-|
   \2/      \2/     \        \4//    \4/     \        \4//    \4/
1tan⁡(y2)+1tan⁡(x2)−(tan⁡2(y4)+1)24(1−tan⁡2(y4))tan⁡(y4)−(tan⁡2(x4)+1)24(1−tan⁡2(x4))tan⁡(x4)\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{\left(\tan^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2}}{4 \left(1 - \tan^{2}{\left(\frac{y}{4} \right)}\right) \tan{\left(\frac{y}{4} \right)}} - \frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \tan{\left(\frac{x}{4} \right)}} 
  1        1        1        1   
------ + ------ - ------ - ------
   /x\      /y\   sin(x)   sin(y)
tan|-|   tan|-|                  
   \2/      \2/
1tan⁡(y2)+1tan⁡(x2)−1sin⁡(y)−1sin⁡(x)\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}} 
                    2                         2                      
       /       2/x\\             /       2/y\\                       
       |1 + cot |-||             |1 + cot |-||                       
       \        \4//             \        \4//           /x\      /y\
- ----------------------- - ----------------------- + cot|-| + cot|-|
    /        2/x\\    /x\     /        2/y\\    /y\      \2/      \2/
  4*|-1 + cot |-||*cot|-|   4*|-1 + cot |-||*cot|-|                  
    \         \4//    \4/     \         \4//    \4/
cot⁡(x2)+cot⁡(y2)−(cot⁡2(y4)+1)24(cot⁡2(y4)−1)cot⁡(y4)−(cot⁡2(x4)+1)24(cot⁡2(x4)−1)cot⁡(x4)\cot{\left(\frac{x}{2} \right)} + \cot{\left(\frac{y}{2} \right)} - \frac{\left(\cot^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{y}{4} \right)} - 1\right) \cot{\left(\frac{y}{4} \right)}} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \cot{\left(\frac{x}{4} \right)}} 
       /x\             /y\         /pi   x\      /pi   y\
    csc|-|          csc|-|      csc|-- - -|   csc|-- - -|
       \2/             \2/         \2    2/      \2    2/
------------- + ------------- - ----------- - -----------
     /pi   x\        /pi   y\          /x\           /y\ 
2*csc|-- - -|   2*csc|-- - -|     2*csc|-|      2*csc|-| 
     \2    2/        \2    2/          \2/           \2/
csc⁡(x2)2csc⁡(−x2+π2)+csc⁡(y2)2csc⁡(−y2+π2)−csc⁡(−y2+π2)2csc⁡(y2)−csc⁡(−x2+π2)2csc⁡(x2)\frac{\csc{\left(\frac{x}{2} \right)}}{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{\csc{\left(\frac{y}{2} \right)}}{2 \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{y}{2} \right)}} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{x}{2} \right)}} 
   cos(x)        cos(y)  
----------- + -----------
   /    pi\      /    pi\
cos|x - --|   cos|y - --|
   \    2 /      \    2 /
cos⁡(x)cos⁡(x−π2)+cos⁡(y)cos⁡(y−π2)\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(y \right)}}{\cos{\left(y - \frac{\pi}{2} \right)}} 
  sin(x)      sin(y)             1                      1          
--------- + --------- - -------------------- - --------------------
     2/x\        2/y\        /x\    /pi   x\        /y\    /pi   y\
2*sin |-|   2*sin |-|   2*sin|-|*sin|-- + -|   2*sin|-|*sin|-- + -|
      \2/         \2/        \2/    \2    2/        \2/    \2    2/
−12sin⁡(y2)sin⁡(y2+π2)+sin⁡(y)2sin⁡2(y2)−12sin⁡(x2)sin⁡(x2+π2)+sin⁡(x)2sin⁡2(x2)- \frac{1}{2 \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{y}{2} + \frac{\pi}{2} \right)}} + \frac{\sin{\left(y \right)}}{2 \sin^{2}{\left(\frac{y}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(\frac{x}{2} \right)}} 
   /x   pi\      /y   pi\          /x\             /y\   
sec|- - --|   sec|- - --|       sec|-|          sec|-|   
   \2   2 /      \2   2 /          \2/             \2/   
----------- + ----------- - ------------- - -------------
       /x\           /y\         /x   pi\        /y   pi\
  2*sec|-|      2*sec|-|    2*sec|- - --|   2*sec|- - --|
       \2/           \2/         \2   2 /        \2   2 /
−sec⁡(x2)2sec⁡(x2−π2)−sec⁡(y2)2sec⁡(y2−π2)+sec⁡(y2−π2)2sec⁡(y2)+sec⁡(x2−π2)2sec⁡(x2)- \frac{\sec{\left(\frac{x}{2} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\sec{\left(\frac{y}{2} \right)}}{2 \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{y}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{x}{2} \right)}} 
   /    pi\    /    pi\
sec|x - --|*sec|y - --|
   \    2 /    \    2 /
-----------------------
       /        pi\    
    sec|x + y - --|    
       \        2 /
sec⁡(x−π2)sec⁡(y−π2)sec⁡(x+y−π2)\frac{\sec{\left(x - \frac{\pi}{2} \right)} \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(x + y - \frac{\pi}{2} \right)}} 
       /        pi\    
    cos|x + y - --|    
       \        2 /    
-----------------------
   /    pi\    /    pi\
cos|x - --|*cos|y - --|
   \    2 /    \    2 /
cos⁡(x+y−π2)cos⁡(x−π2)cos⁡(y−π2)\frac{\cos{\left(x + y - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} \cos{\left(y - \frac{\pi}{2} \right)}} 
    1        1        sin(x)       sin(y)  
- ------ - ------ + ---------- + ----------
  sin(x)   sin(y)   1 - cos(x)   1 - cos(y)
−1sin⁡(y)−1sin⁡(x)+sin⁡(y)1−cos⁡(y)+sin⁡(x)1−cos⁡(x)- \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}} + \frac{\sin{\left(y \right)}}{1 - \cos{\left(y \right)}} + \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}} 
 sin(2*x)    sin(2*y)
--------- + ---------
     2           2   
2*sin (x)   2*sin (y)
sin⁡(2y)2sin⁡2(y)+sin⁡(2x)2sin⁡2(x)\frac{\sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} 
  1        1             1                 1       
------ + ------ - --------------- - ---------------
   /x\      /y\        /x\    /x\        /y\    /y\
tan|-|   tan|-|   2*cos|-|*sin|-|   2*cos|-|*sin|-|
   \2/      \2/        \2/    \2/        \2/    \2/
1tan⁡(y2)+1tan⁡(x2)−12sin⁡(y2)cos⁡(y2)−12sin⁡(x2)cos⁡(x2)\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{y}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} 
  1        1   
------ + ------
tan(x)   tan(y)
1tan⁡(y)+1tan⁡(x)\frac{1}{\tan{\left(y \right)}} + \frac{1}{\tan{\left(x \right)}} 
/       2/x\\ /       2/y\\    /x   y\
|1 + cot |-||*|1 + cot |-||*cot|- + -|
\        \2// \        \2//    \2   2/
--------------------------------------
    /       2/x   y\\    /x\    /y\   
  2*|1 + cot |- + -||*cot|-|*cot|-|   
    \        \2   2//    \2/    \2/
(cot⁡2(x2)+1)(cot⁡2(y2)+1)cot⁡(x2+y2)2(cot⁡2(x2+y2)+1)cot⁡(x2)cot⁡(y2)\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{y}{2} \right)}} 
/       2/x\\ /       2/y\\    /x   y\
|1 + tan |-||*|1 + tan |-||*tan|- + -|
\        \2// \        \2//    \2   2/
--------------------------------------
    /       2/x   y\\    /x\    /y\   
  2*|1 + tan |- + -||*tan|-|*tan|-|   
    \        \2   2//    \2/    \2/
(tan⁡2(x2)+1)(tan⁡2(y2)+1)tan⁡(x2+y2)2(tan⁡2(x2+y2)+1)tan⁡(x2)tan⁡(y2)\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)}} 
                                     2/x\      2/y\
                                  sin |-|   sin |-|
    sin(x)           sin(y)           \2/       \2/
-------------- + -------------- - ------- - -------
2*(1 - cos(x))   2*(1 - cos(y))    sin(x)    sin(y)
−sin⁡2(x2)sin⁡(x)−sin⁡2(y2)sin⁡(y)+sin⁡(y)2(1−cos⁡(y))+sin⁡(x)2(1−cos⁡(x))- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin^{2}{\left(\frac{y}{2} \right)}}{\sin{\left(y \right)}} + \frac{\sin{\left(y \right)}}{2 \left(1 - \cos{\left(y \right)}\right)} + \frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)} 
   /x   pi\      /y   pi\      /x\    /x   pi\      /y\    /y   pi\
sec|- - --|   sec|- - --|   sec|-|*sec|- - --|   sec|-|*sec|- - --|
   \2   2 /      \2   2 /      \2/    \2   2 /      \2/    \2   2 /
----------- + ----------- - ------------------ - ------------------
      /x\           /y\             2                    2         
   sec|-|        sec|-|                                            
      \2/           \2/
−sec⁡(x2)sec⁡(x2−π2)2−sec⁡(y2)sec⁡(y2−π2)2+sec⁡(y2−π2)sec⁡(y2)+sec⁡(x2−π2)sec⁡(x2)- \frac{\sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2} - \frac{\sec{\left(\frac{y}{2} \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2} + \frac{\sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{y}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{x}{2} \right)}} 
      /x\           /y\        /x\    /pi   x\      /y\    /pi   y\
   csc|-|        csc|-|     csc|-|*csc|-- - -|   csc|-|*csc|-- - -|
      \2/           \2/        \2/    \2    2/      \2/    \2    2/
----------- + ----------- - ------------------ - ------------------
   /pi   x\      /pi   y\           2                    2         
csc|-- - -|   csc|-- - -|                                          
   \2    2/      \2    2/
−csc⁡(x2)csc⁡(−x2+π2)2+csc⁡(x2)csc⁡(−x2+π2)−csc⁡(y2)csc⁡(−y2+π2)2+csc⁡(y2)csc⁡(−y2+π2)- \frac{\csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\csc{\left(\frac{x}{2} \right)}}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(\frac{y}{2} \right)} \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\csc{\left(\frac{y}{2} \right)}}{\csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} 
    1        1         /x\      /y\
- ------ - ------ + cot|-| + cot|-|
  sin(x)   sin(y)      \2/      \2/
cot⁡(x2)+cot⁡(y2)−1sin⁡(y)−1sin⁡(x)\cot{\left(\frac{x}{2} \right)} + \cot{\left(\frac{y}{2} \right)} - \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}} 
     2/x\      2/y\                        
  sin |-|   sin |-|                        
      \2/       \2/     sin(x)      sin(y) 
- ------- - ------- + --------- + ---------
   sin(x)    sin(y)        2/x\        2/y\
                      4*sin |-|   4*sin |-|
                            \2/         \2/
−sin⁡2(x2)sin⁡(x)−sin⁡2(y2)sin⁡(y)+sin⁡(y)4sin⁡2(y2)+sin⁡(x)4sin⁡2(x2)- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin^{2}{\left(\frac{y}{2} \right)}}{\sin{\left(y \right)}} + \frac{\sin{\left(y \right)}}{4 \sin^{2}{\left(\frac{y}{2} \right)}} + \frac{\sin{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}} 
      /x\           /y\                                                
   cos|-|        cos|-|                                                
      \2/           \2/              1                      1          
----------- + ----------- - -------------------- - --------------------
   /x   pi\      /y   pi\        /x\    /x   pi\        /y\    /y   pi\
cos|- - --|   cos|- - --|   2*cos|-|*cos|- - --|   2*cos|-|*cos|- - --|
   \2   2 /      \2   2 /        \2/    \2   2 /        \2/    \2   2 /
cos⁡(x2)cos⁡(x2−π2)+cos⁡(y2)cos⁡(y2−π2)−12cos⁡(y2)cos⁡(y2−π2)−12cos⁡(x2)cos⁡(x2−π2)\frac{\cos{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{y}{2} \right)}}{\cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{1}{2 \cos{\left(\frac{y}{2} \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{1}{2 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} 
   csc(x)        csc(y)  
----------- + -----------
   /pi    \      /pi    \
csc|-- - x|   csc|-- - y|
   \2     /      \2     /
csc⁡(x)csc⁡(−x+π2)+csc⁡(y)csc⁡(−y+π2)\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(y \right)}}{\csc{\left(- y + \frac{\pi}{2} \right)}} 
                         /x\      /y\
                      tan|-|   tan|-|
   1          1          \2/      \2/
-------- + -------- - ------ - ------
     /x\        /y\     2        2   
2*tan|-|   2*tan|-|                  
     \2/        \2/
−tan⁡(x2)2−tan⁡(y2)2+12tan⁡(y2)+12tan⁡(x2)- \frac{\tan{\left(\frac{x}{2} \right)}}{2} - \frac{\tan{\left(\frac{y}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{y}{2} \right)}} + \frac{1}{2 \tan{\left(\frac{x}{2} \right)}} 
   /x\      /y\                      
cot|-|   cot|-|                      
   \2/      \2/      1          1    
------ + ------ - -------- - --------
  2        2           /x\        /y\
                  2*cot|-|   2*cot|-|
                       \2/        \2/
cot⁡(x2)2+cot⁡(y2)2−12cot⁡(y2)−12cot⁡(x2)\frac{\cot{\left(\frac{x}{2} \right)}}{2} + \frac{\cot{\left(\frac{y}{2} \right)}}{2} - \frac{1}{2 \cot{\left(\frac{y}{2} \right)}} - \frac{1}{2 \cot{\left(\frac{x}{2} \right)}} 
cot(x/2)/2 + cot(y/2)/2 - 1/(2*cot(x/2)) - 1/(2*cot(y/2))
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