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

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

Expresión a simplificar:

Solución

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