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

Expresión a simplificar:

Solución

Ha introducido [src]
          2    
 1 - 2*cos (x) 
---------------
sin(x) - cos(x)
$$\frac{1 - 2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$
(1 - 2*cos(x)^2)/(sin(x) - cos(x))
Simplificación general [src]
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*cos|x + --| 
     \    4 / 
$$\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}}$$
sqrt(2)*cos(2*x)/(2*cos(x + pi/4))
Potencias [src]
                           2       
             / I*x    -I*x\        
             |e      e    |        
       1 - 2*|---- + -----|        
             \ 2       2  /        
-----------------------------------
   I*x    -I*x     /   -I*x    I*x\
  e      e       I*\- e     + e   /
- ---- - ----- - ------------------
   2       2             2         
$$\frac{1 - 2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} - \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}}$$
(1 - 2*(exp(i*x)/2 + exp(-i*x)/2)^2)/(-exp(i*x)/2 - exp(-i*x)/2 - i*(-exp(-i*x) + exp(i*x))/2)
Combinatoria [src]
           2    
 -1 + 2*cos (x) 
----------------
-sin(x) + cos(x)
$$\frac{2 \cos^{2}{\left(x \right)} - 1}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
(-1 + 2*cos(x)^2)/(-sin(x) + cos(x))
Denominador común [src]
                                 2    
                       -1 + 2*sin (x) 
2*cos(x) + 2*sin(x) + ----------------
                      -sin(x) + cos(x)
$$2 \sin{\left(x \right)} + 2 \cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)} - 1}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
2*cos(x) + 2*sin(x) + (-1 + 2*sin(x)^2)/(-sin(x) + cos(x))
Abrimos la expresión [src]
                          2      
       1             2*cos (x)   
--------------- - ---------------
sin(x) - cos(x)   sin(x) - cos(x)
$$- \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$
1/(sin(x) - cos(x)) - 2*cos(x)^2/(sin(x) - cos(x))
Parte trigonométrica [src]
  ___    2    /       2/x   pi\\ /       2   \
\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
              \        \2   8 //              
----------------------------------------------
               /       2/x   pi\\             
             2*|1 - tan |- + --||             
               \        \2   8 //             
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}$$
           2        
  1 - ------------  
         2/pi    \  
      csc |-- - x|  
          \2     /  
--------------------
  1           1     
------ - -----------
csc(x)      /pi    \
         csc|-- - x|
            \2     /
$$\frac{1 - \frac{2}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}}{- \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}}$$
  ___    /pi      \
\/ 2 *sin|-- + 2*x|
         \2       /
-------------------
       /    3*pi\  
  2*sin|x + ----|  
       \     4  /  
$$\frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{3 \pi}{4} \right)}}$$
  ___    /     pi\
\/ 2 *csc|-x + --|
         \     4 /
------------------
      /pi      \  
 2*csc|-- - 2*x|  
      \2       /  
$$\frac{\sqrt{2} \csc{\left(- x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
  ___ /       2/x   pi\\ /        2   \
\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
      \        \2   8 //               
---------------------------------------
    /       2   \ /        2/x   pi\\  
  2*\1 + cot (x)/*|-1 + cot |- + --||  
                  \         \2   8 //  
$$\frac{\sqrt{2} \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}$$
  ___ /       2/x   pi\\ /       2   \
\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
      \        \2   8 //              
--------------------------------------
    /       2   \ /       2/x   pi\\  
  2*\1 + tan (x)/*|1 - tan |- + --||  
                  \        \2   8 //  
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
           2        
    1 - -------     
           2        
        sec (x)     
--------------------
     1          1   
----------- - ------
   /    pi\   sec(x)
sec|x - --|         
   \    2 /         
$$\frac{1 - \frac{2}{\sec^{2}{\left(x \right)}}}{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}}$$
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*cos|x + --| 
     \    4 / 
$$\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}}$$
           2/    pi\  
  1 - 2*sin |x + --|  
            \    2 /  
----------------------
     /    pi\         
- sin|x + --| + sin(x)
     \    2 /         
$$\frac{1 - 2 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} - \sin{\left(x + \frac{\pi}{2} \right)}}$$
  ___             /     pi\
\/ 2 *cos(2*x)*csc|-x + --|
                  \     4 /
---------------------------
             2             
$$\frac{\sqrt{2} \cos{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{4} \right)}}{2}$$
         2     
  1 - -------  
         2     
      sec (x)  
---------------
  1        1   
------ - ------
csc(x)   sec(x)
$$\frac{1 - \frac{2}{\sec^{2}{\left(x \right)}}}{- \frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}}$$
  ___             /    pi\
\/ 2 *cos(2*x)*sec|x + --|
                  \    4 /
--------------------------
            2             
$$\frac{\sqrt{2} \cos{\left(2 x \right)} \sec{\left(x + \frac{\pi}{4} \right)}}{2}$$
                      ___                            
                   -\/ 2 *cos(2*x)                   
-----------------------------------------------------
                    ___________    ___________       
    ___            /       ___    /       ___        
- \/ 2 *cos(x) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(x)
$$- \frac{\sqrt{2} \cos{\left(2 x \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(x \right)} - \sqrt{2} \cos{\left(x \right)}}$$
                       2   
          /       2/x\\    
        2*|1 - tan |-||    
          \        \2//    
    1 - ----------------   
                      2    
         /       2/x\\     
         |1 + tan |-||     
         \        \2//     
---------------------------
         2/x\          /x\ 
  1 - tan |-|     2*tan|-| 
          \2/          \2/ 
- ----------- + -----------
         2/x\          2/x\
  1 + tan |-|   1 + tan |-|
          \2/           \2/
$$\frac{- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}{- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}$$
             2       
    1 - 2*cos (x)    
---------------------
             /    pi\
-cos(x) + cos|x - --|
             \    2 /
$$\frac{1 - 2 \cos^{2}{\left(x \right)}}{- \cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}$$
                       2    
         /        2/x\\     
       2*|-1 + cot |-||     
         \         \2//     
   1 - -----------------    
                      2     
         /       2/x\\      
         |1 + cot |-||      
         \        \2//      
----------------------------
          2/x\          /x\ 
  -1 + cot |-|     2*cot|-| 
           \2/          \2/ 
- ------------ + -----------
         2/x\           2/x\
  1 + cot |-|    1 + cot |-|
          \2/            \2/
$$\frac{- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}{- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}$$
  ___    /    pi\
\/ 2 *sec|x + --|
         \    4 /
-----------------
    2*sec(2*x)   
$$\frac{\sqrt{2} \sec{\left(x + \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}}$$
sqrt(2)*sec(x + pi/4)/(2*sec(2*x))
Respuesta numérica [src]
(1.0 - 2.0*cos(x)^2)/(-cos(x) + sin(x))
(1.0 - 2.0*cos(x)^2)/(-cos(x) + sin(x))