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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x)/(1-cos(x))-sin(x)^2/(1-cos(x))^2

¿Cómo vas a descomponer esta cos(x)/(1-cos(x))-sin(x)^2/(1-cos(x))^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
−sin⁡2(x)(1−cos⁡(x))2+cos⁡(x)1−cos⁡(x)- \frac{\sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} 
cos(x)/(1 - cos(x)) - sin(x)^2/(1 - cos(x))^2
Simplificación general [src] 
     1     
-----------
-1 + cos(x)
1cos⁡(x)−1\frac{1}{\cos{\left(x \right)} - 1} 
1/(-1 + cos(x))
Respuesta numérica [src] 
cos(x)/(1.0 - cos(x)) - sin(x)^2/(1.0 - cos(x))^2
cos(x)/(1.0 - cos(x)) - sin(x)^2/(1.0 - cos(x))^2
Denominador común [src] 
             2             
     -1 + sin (x) + cos(x) 
-1 - ----------------------
            2              
     1 + cos (x) - 2*cos(x)
−sin⁡2(x)+cos⁡(x)−1cos⁡2(x)−2cos⁡(x)+1−1- \frac{\sin^{2}{\left(x \right)} + \cos{\left(x \right)} - 1}{\cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 1} - 1 
-1 - (-1 + sin(x)^2 + cos(x))/(1 + cos(x)^2 - 2*cos(x))
Denominador racional [src] 
            2             2                
(1 - cos(x)) *cos(x) - sin (x)*(1 - cos(x))
-------------------------------------------
                           3               
               (1 - cos(x))
(1−cos⁡(x))2cos⁡(x)−(1−cos⁡(x))sin⁡2(x)(1−cos⁡(x))3\frac{\left(1 - \cos{\left(x \right)}\right)^{2} \cos{\left(x \right)} - \left(1 - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{3}} 
((1 - cos(x))^2*cos(x) - sin(x)^2*(1 - cos(x)))/(1 - cos(x))^3
Potencias [src] 
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
cos⁡(x)1−cos⁡(x)−sin⁡2(x)(1−cos⁡(x))2\frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
   I*x    -I*x                          
  e      e                           2  
  ---- + -----       /   -I*x    I*x\   
   2       2         \- e     + e   /   
---------------- + ---------------------
     I*x    -I*x                       2
    e      e         /     I*x    -I*x\ 
1 - ---- - -----     |    e      e    | 
     2       2     4*|1 - ---- - -----| 
                     \     2       2  /
eix2+e−ix2−eix2+1−e−ix2+(eix−e−ix)24(−eix2+1−e−ix2)2\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}} + \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)^{2}} 
(exp(i*x)/2 + exp(-i*x)/2)/(1 - exp(i*x)/2 - exp(-i*x)/2) + (-exp(-i*x) + exp(i*x))^2/(4*(1 - exp(i*x)/2 - exp(-i*x)/2)^2)
Abrimos la expresión [src] 
                       2           
  cos(x)            sin (x)        
---------- - ----------------------
1 - cos(x)          2              
             1 + cos (x) - 2*cos(x)
−sin⁡2(x)cos⁡2(x)−2cos⁡(x)+1+cos⁡(x)1−cos⁡(x)- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} - 2 \cos{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} 
cos(x)/(1 - cos(x)) - sin(x)^2/(1 + cos(x)^2 - 2*cos(x))
Unión de expresiones racionales [src] 
     2                         
- sin (x) + (1 - cos(x))*cos(x)
-------------------------------
                     2         
         (1 - cos(x))
(1−cos⁡(x))cos⁡(x)−sin⁡2(x)(1−cos⁡(x))2\frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)} - \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
(-sin(x)^2 + (1 - cos(x))*cos(x))/(1 - cos(x))^2
Compilar la expresión [src] 
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
cos⁡(x)1−cos⁡(x)−sin⁡2(x)(1−cos⁡(x))2\frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
cos(x)/(1 - cos(x)) - sin(x)^2/(1 - cos(x))^2
Parte trigonométrica [src] 
                   2      
  cos(x)        sin (x)   
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
cos⁡(x)1−cos⁡(x)−sin⁡2(x)(1−cos⁡(x))2\frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
                  2/x\                              2/x\             
          -1 + cot |-|                         4*cot |-|             
                   \2/                               \2/             
-------------------------------- - ----------------------------------
              /            2/x\\                                    2
              |    -1 + cot |-||                  /            2/x\\ 
/       2/x\\ |             \2/|                2 |    -1 + cot |-|| 
|1 + cot |-||*|1 - ------------|   /       2/x\\  |             \2/| 
\        \2// |           2/x\ |   |1 + cot |-|| *|1 - ------------| 
              |    1 + cot |-| |   \        \2//  |           2/x\ | 
              \            \2/ /                  |    1 + cot |-| | 
                                                  \            \2/ /
cot⁡2(x2)−1(−cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1)−4cot⁡2(x2)(−cot⁡2(x2)−1cot⁡2(x2)+1+1)2(cot⁡2(x2)+1)2\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{4 \cot^{2}{\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)^{2}} 
       1        
----------------
            2/x\
     1 - tan |-|
             \2/
-1 + -----------
            2/x\
     1 + tan |-|
             \2/
11−tan⁡2(x2)tan⁡2(x2)+1−1\frac{1}{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1} 
         1                        1             
------------------- - --------------------------
/      1   \                      2             
|1 - ------|*sec(x)   /      1   \     2/    pi\
\    sec(x)/          |1 - ------| *sec |x - --|
                      \    sec(x)/      \    2 /
1(1−1sec⁡(x))sec⁡(x)−1(1−1sec⁡(x))2sec⁡2(x−π2)\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
     1     
-----------
       1   
-1 + ------
     sec(x)
1−1+1sec⁡(x)\frac{1}{-1 + \frac{1}{\sec{\left(x \right)}}} 
     /    pi\                       
  sin|x + --|             2         
     \    2 /          sin (x)      
--------------- - ------------------
       /    pi\                    2
1 - sin|x + --|   /       /    pi\\ 
       \    2 /   |1 - sin|x + --|| 
                  \       \    2 //
sin⁡(x+π2)1−sin⁡(x+π2)−sin⁡2(x)(1−sin⁡(x+π2))2\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{1 - \sin{\left(x + \frac{\pi}{2} \right)}} - \frac{\sin^{2}{\left(x \right)}}{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} 
                 2/    pi\
              cos |x - --|
  cos(x)          \    2 /
---------- - -------------
1 - cos(x)               2
             (1 - cos(x))
cos⁡(x)1−cos⁡(x)−cos⁡2(x−π2)(1−cos⁡(x))2\frac{\cos{\left(x \right)}}{1 - \cos{\left(x \right)}} - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} 
        1        
-----------------
             2/x\
     -1 + cot |-|
              \2/
-1 + ------------
            2/x\ 
     1 + cot |-| 
             \2/
1cot⁡2(x2)−1cot⁡2(x2)+1−1\frac{1}{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1} 
     1     
-----------
-1 + cos(x)
1cos⁡(x)−1\frac{1}{\cos{\left(x \right)} - 1} 
       1        
----------------
          1     
-1 + -----------
        /pi    \
     csc|-- - x|
        \2     /
1−1+1csc⁡(−x+π2)\frac{1}{-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} 
              1                             1             
----------------------------- - --------------------------
/         1     \    /pi    \                    2        
|1 - -----------|*csc|-- - x|   /         1     \     2   
|       /pi    \|    \2     /   |1 - -----------| *csc (x)
|    csc|-- - x||               |       /pi    \|         
\       \2     //               |    csc|-- - x||         
                                \       \2     //
1(1−1csc⁡(−x+π2))csc⁡(−x+π2)−1(1−1csc⁡(−x+π2))2csc⁡2(x)\frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc^{2}{\left(x \right)}} 
                 2/x\                              2/x\            
          1 - tan |-|                         4*tan |-|            
                  \2/                               \2/            
------------------------------- - ---------------------------------
              /           2/x\\                                   2
              |    1 - tan |-||                  /           2/x\\ 
/       2/x\\ |            \2/|                2 |    1 - tan |-|| 
|1 + tan |-||*|1 - -----------|   /       2/x\\  |            \2/| 
\        \2// |           2/x\|   |1 + tan |-|| *|1 - -----------| 
              |    1 + tan |-||   \        \2//  |           2/x\| 
              \            \2//                  |    1 + tan |-|| 
                                                 \            \2//
1−tan⁡2(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1)−4tan⁡2(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+1)2(tan⁡2(x2)+1)2\frac{1 - \tan^{2}{\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) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{4 \tan^{2}{\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}} 
         1                      1          
------------------- - ---------------------
/      1   \                      2        
|1 - ------|*sec(x)   /      1   \     2   
\    sec(x)/          |1 - ------| *csc (x)
                      \    sec(x)/
1(1−1sec⁡(x))sec⁡(x)−1(1−1sec⁡(x))2csc⁡2(x)\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc^{2}{\left(x \right)}} 
       1        
----------------
        /    pi\
-1 + sin|x + --|
        \    2 /
1sin⁡(x+π2)−1\frac{1}{\sin{\left(x + \frac{\pi}{2} \right)} - 1} 
1/(-1 + sin(x + pi/2))
Combinatoria [src] 
 /   2         2            \ 
-\cos (x) + sin (x) - cos(x)/ 
------------------------------
                     2        
        (-1 + cos(x))
−sin⁡2(x)+cos⁡2(x)−cos⁡(x)(cos⁡(x)−1)2- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}} 
-(cos(x)^2 + sin(x)^2 - cos(x))/(-1 + cos(x))^2
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