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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    cos(2*x)   
---------------
cos(x) - sin(x)
cos⁡(2x)−sin⁡(x)+cos⁡(x)\frac{\cos{\left(2 x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}} 
cos(2*x)/(cos(x) - sin(x))
Simplificación general [src] 
  ___         
\/ 2 *cos(2*x)
--------------
     /    pi\ 
2*cos|x + --| 
     \    4 /
2cos⁡(2x)2cos⁡(x+π4)\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    2*I*x        
         e         e             
         ------- + ------        
            2        2           
---------------------------------
 I*x    -I*x     /   -I*x    I*x\
e      e       I*\- e     + e   /
---- + ----- + ------------------
 2       2             2
e2ix2+e−2ix2i(eix−e−ix)2+eix2+e−ix2\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)
Abrimos la expresión [src] 
                            2      
         1             2*cos (x)   
- --------------- + ---------------
  cos(x) - sin(x)   cos(x) - sin(x)
2cos⁡2(x)−sin⁡(x)+cos⁡(x)−1−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/(cos(x) - sin(x)) + 2*cos(x)^2/(cos(x) - sin(x))
Respuesta numérica [src] 
cos(2*x)/(-sin(x) + cos(x))
cos(2*x)/(-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 //
2(1−tan⁡2(x))(tan⁡2(x2+π8)+1)cos⁡2(x)2(1−tan⁡2(x2+π8))\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/x   pi\\ /        2   \
\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
      \        \2   8 //               
---------------------------------------
    /       2   \ /        2/x   pi\\  
  2*\1 + cot (x)/*|-1 + cot |- + --||  
                  \         \2   8 //
2(cot⁡2(x)−1)(cot⁡2(x2+π8)+1)2(cot⁡2(x)+1)(cot⁡2(x2+π8)−1)\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)} 
    ___    /x   pi\
2*\/ 2 *tan|- + --|
           \2   8 /
-------------------
         2/x   pi\ 
  1 + tan |- + --| 
          \2   8 /
22tan⁡(x2+π8)tan⁡2(x2+π8)+1\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 *cos(2*x)
--------------
     /    pi\ 
2*cos|x + --| 
     \    4 /
2cos⁡(2x)2cos⁡(x+π4)\frac{\sqrt{2} \cos{\left(2 x \right)}}{2 \cos{\left(x + \frac{\pi}{4} \right)}} 
  ___             /    pi\
\/ 2 *cos(2*x)*sec|x + --|
                  \    4 /
--------------------------
            2
2cos⁡(2x)sec⁡(x+π4)2\frac{\sqrt{2} \cos{\left(2 x \right)} \sec{\left(x + \frac{\pi}{4} \right)}}{2} 
       /pi      \    
    sin|-- + 2*x|    
       \2       /    
---------------------
             /    pi\
-sin(x) + sin|x + --|
             \    2 /
sin⁡(2x+π2)−sin⁡(x)+sin⁡(x+π2)\frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{- \sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}} 
    ___    2/x   pi\    /x   pi\
2*\/ 2 *sin |- + --|*cot|- + --|
            \2   8 /    \2   8 /
22sin⁡2(x2+π8)cot⁡(x2+π8)2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)} 
    ___    /x   pi\
2*\/ 2 *cot|- + --|
           \2   8 /
-------------------
         2/x   pi\ 
  1 + cot |- + --| 
          \2   8 /
22cot⁡(x2+π8)cot⁡2(x2+π8)+1\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                  
               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//
1−tan⁡2(x)(1−tan⁡2(x2)tan⁡2(x2)+1−2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x)+1)\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)} 
  ___ /       2/x   pi\\ /       2   \
\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
      \        \2   8 //              
--------------------------------------
    /       2   \ /       2/x   pi\\  
  2*\1 + tan (x)/*|1 - tan |- + --||  
                  \        \2   8 //
2(1−tan⁡2(x))(tan⁡2(x2+π8)+1)2(1−tan⁡2(x2+π8))(tan⁡2(x)+1)\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 *cos(2*x)                   
-----------------------------------------------------
                    ___________    ___________       
    ___            /       ___    /       ___        
- \/ 2 *cos(x) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(x)
−2cos⁡(2x)2−22+2sin⁡(x)−2cos⁡(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)}} 
       cos(2*x)       
----------------------
     /    pi\         
- cos|x - --| + cos(x)
     \    2 /
cos⁡(2x)cos⁡(x)−cos⁡(x−π2)\frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)} - \cos{\left(x - \frac{\pi}{2} \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//
cot⁡2(x)−1(cot⁡2(x2)−1cot⁡2(x2)+1−2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x)+1)\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)} 
            1             
--------------------------
/  1        1   \         
|------ - ------|*sec(2*x)
\sec(x)   csc(x)/
1(1sec⁡(x)−1csc⁡(x))sec⁡(2x)\frac{1}{\left(\frac{1}{\sec{\left(x \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
  ___    /pi      \
\/ 2 *sin|-- + 2*x|
         \2       /
-------------------
       /    3*pi\  
  2*sin|x + ----|  
       \     4  /
2sin⁡(2x+π2)2sin⁡(x+3π4)\frac{\sqrt{2} \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{3 \pi}{4} \right)}} 
               1               
-------------------------------
/  1           1     \         
|------ - -----------|*sec(2*x)
|sec(x)      /    pi\|         
|         sec|x - --||         
\            \    2 //
1(−1sec⁡(x−π2)+1sec⁡(x))sec⁡(2x)\frac{1}{\left(- \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}} 
  ___    /    pi\
\/ 2 *cos|x - --|
         \    4 /
2cos⁡(x−π4)\sqrt{2} \cos{\left(x - \frac{\pi}{4} \right)} 
     ___   
   \/ 2    
-----------
   /    pi\
sec|x - --|
   \    4 /
2sec⁡(x−π4)\frac{\sqrt{2}}{\sec{\left(x - \frac{\pi}{4} \right)}} 
                 1                  
------------------------------------
/     1          1   \    /pi      \
|----------- - ------|*csc|-- - 2*x|
|   /pi    \   csc(x)|    \2       /
|csc|-- - x|         |              
\   \2     /         /
1(1csc⁡(−x+π2)−1csc⁡(x))csc⁡(−2x+π2)\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)}} 
  ___    /     pi\
\/ 2 *csc|-x + --|
         \     4 /
------------------
      /pi      \  
 2*csc|-- - 2*x|  
      \2       /
2csc⁡(−x+π4)2csc⁡(−2x+π2)\frac{\sqrt{2} \csc{\left(- x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
  ___             /     pi\
\/ 2 *cos(2*x)*csc|-x + --|
                  \     4 /
---------------------------
             2
2cos⁡(2x)csc⁡(−x+π4)2\frac{\sqrt{2} \cos{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{4} \right)}}{2} 
     ___   
   \/ 2    
-----------
   /    pi\
csc|x + --|
   \    4 /
2csc⁡(x+π4)\frac{\sqrt{2}}{\csc{\left(x + \frac{\pi}{4} \right)}} 
  ___    /    pi\
\/ 2 *sec|x + --|
         \    4 /
-----------------
    2*sec(2*x)
2sec⁡(x+π4)2sec⁡(2x)\frac{\sqrt{2} \sec{\left(x + \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}} 
  ___    /    pi\
\/ 2 *sin|x + --|
         \    4 /
2sin⁡(x+π4)\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} 
sqrt(2)*sin(x + pi/4)
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