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Simplificación de expresiones/ Descomposición en fracciones simples/ sinx/(1-cosx)^2

¿Cómo vas a descomponer esta sinx/(1-cosx)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    sin(x)   
-------------
            2
(1 - cos(x))
sin⁡(x)(1−cos⁡(x))2\frac{\sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
sin(x)/(1 - cos(x))^2
Simplificación general [src] 
    sin(x)    
--------------
             2
(-1 + cos(x))
sin⁡(x)(cos⁡(x)−1)2\frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} 
sin(x)/(-1 + cos(x))^2
Potencias [src] 
    /   -I*x    I*x\ 
 -I*\- e     + e   / 
---------------------
                    2
  /     I*x    -I*x\ 
  |    e      e    | 
2*|1 - ---- - -----| 
  \     2       2  /
−i(eix−e−ix)2(−eix2+1−e−ix2)2- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)^{2}} 
-i*(-exp(-i*x) + exp(i*x))/(2*(1 - exp(i*x)/2 - exp(-i*x)/2)^2)
Combinatoria [src] 
    sin(x)    
--------------
             2
(-1 + cos(x))
sin⁡(x)(cos⁡(x)−1)2\frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} 
sin(x)/(-1 + cos(x))^2
Abrimos la expresión [src] 
        sin(x)        
----------------------
       2              
1 + cos (x) - 2*cos(x)
sin⁡(x)cos⁡2(x)−2cos⁡(x)+1\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 1} 
sin(x)/(1 + cos(x)^2 - 2*cos(x))
Parte trigonométrica [src] 
            1             
--------------------------
                  2       
/          1     \        
|-1 + -----------| *csc(x)
|        /pi    \|        
|     csc|-- - x||        
\        \2     //
1(−1+1csc⁡(−x+π2))2csc⁡(x)\frac{1}{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} 
       sin(x)      
-------------------
                  2
/        /    pi\\ 
|-1 + sin|x + --|| 
\        \    2 //
sin⁡(x)(sin⁡(x+π2)−1)2\frac{\sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} - 1\right)^{2}} 
            1            
-------------------------
            2            
/      1   \     /    pi\
|1 - ------| *sec|x - --|
\    sec(x)/     \    2 /
1(1−1sec⁡(x))2sec⁡(x−π2)\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} 
                  /x\             
             2*cot|-|             
                  \2/             
----------------------------------
                                 2
              /             2/x\\ 
              |     -1 + cot |-|| 
/       2/x\\ |              \2/| 
|1 + cot |-||*|-1 + ------------| 
\        \2// |            2/x\ | 
              |     1 + cot |-| | 
              \             \2/ /
2cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1−1)2(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  /x\            
             2*cot|-|            
                  \2/            
---------------------------------
                                2
              /            2/x\\ 
              |    -1 + cot |-|| 
/       2/x\\ |             \2/| 
|1 + cot |-||*|1 - ------------| 
\        \2// |           2/x\ | 
              |    1 + cot |-| | 
              \            \2/ /
2cot⁡(x2)(−cot⁡2(x2)−1cot⁡2(x2)+1+1)2(cot⁡2(x2)+1)\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
            1             
--------------------------
             2            
/       1   \     /    pi\
|-1 + ------| *sec|x - --|
\     sec(x)/     \    2 /
1(−1+1sec⁡(x))2sec⁡(x−π2)\frac{1}{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} 
    /    pi\  
 cos|x - --|  
    \    2 /  
--------------
             2
(-1 + cos(x))
cos⁡(x−π2)(cos⁡(x)−1)2\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} 
    sin(x)    
--------------
             2
(-1 + cos(x))
sin⁡(x)(cos⁡(x)−1)2\frac{\sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} 
      sin(x)      
------------------
                 2
/       /    pi\\ 
|1 - sin|x + --|| 
\       \    2 //
sin⁡(x)(1−sin⁡(x+π2))2\frac{\sin{\left(x \right)}}{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} 
    /    pi\ 
 cos|x - --| 
    \    2 / 
-------------
            2
(1 - cos(x))
cos⁡(x−π2)(1−cos⁡(x))2\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
            1            
-------------------------
                 2       
/         1     \        
|1 - -----------| *csc(x)
|       /pi    \|        
|    csc|-- - x||        
\       \2     //
1(1−1csc⁡(−x+π2))2csc⁡(x)\frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} 
                 /x\            
            2*tan|-|            
                 \2/            
--------------------------------
                               2
              /           2/x\\ 
              |    1 - tan |-|| 
/       2/x\\ |            \2/| 
|1 + tan |-||*|1 - -----------| 
\        \2// |           2/x\| 
              |    1 + tan |-|| 
              \            \2//
2tan⁡(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+1)2(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
         1          
--------------------
            2       
/      1   \        
|1 - ------| *csc(x)
\    sec(x)/
1(1−1sec⁡(x))2csc⁡(x)\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} 
                  /x\            
             2*tan|-|            
                  \2/            
---------------------------------
                                2
              /            2/x\\ 
              |     1 - tan |-|| 
/       2/x\\ |             \2/| 
|1 + tan |-||*|-1 + -----------| 
\        \2// |            2/x\| 
              |     1 + tan |-|| 
              \             \2//
2tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1−1)2(tan⁡2(x2)+1)\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
2*tan(x/2)/((1 + tan(x/2)^2)*(-1 + (1 - tan(x/2)^2)/(1 + tan(x/2)^2))^2)
Respuesta numérica [src] 
sin(x)/(1.0 - cos(x))^2
sin(x)/(1.0 - cos(x))^2
Denominador común [src] 
        sin(x)        
----------------------
       2              
1 + cos (x) - 2*cos(x)
sin⁡(x)cos⁡2(x)−2cos⁡(x)+1\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 1} 
sin(x)/(1 + cos(x)^2 - 2*cos(x))
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