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Simplificación de expresiones/ Descomposición en fracciones simples/ cos2x-sin^2/(2sin^2-cosx)

¿Cómo vas a descomponer esta cos2x-sin^2/(2sin^2-cosx) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                   2         
                sin (x)      
cos(2*x) - ------------------
                2            
           2*sin (x) - cos(x)
cos⁡(2x)−sin⁡2(x)2sin⁡2(x)−cos⁡(x)\cos{\left(2 x \right)} - \frac{\sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}} 
cos(2*x) - sin(x)^2/(2*sin(x)^2 - cos(x))
Simplificación general [src] 
     2      /               2   \         
- sin (x) + \-cos(x) + 2*sin (x)/*cos(2*x)
------------------------------------------
                          2               
           -cos(x) + 2*sin (x)
(2sin⁡2(x)−cos⁡(x))cos⁡(2x)−sin⁡2(x)2sin⁡2(x)−cos⁡(x)\frac{\left(2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(2 x \right)} - \sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}} 
(-sin(x)^2 + (-cos(x) + 2*sin(x)^2)*cos(2*x))/(-cos(x) + 2*sin(x)^2)
Respuesta numérica [src] 
-sin(x)^2/(-cos(x) + 2.0*sin(x)^2) + cos(2*x)
-sin(x)^2/(-cos(x) + 2.0*sin(x)^2) + cos(2*x)
Denominador racional [src] 
     2      /               2   \         
- sin (x) + \-cos(x) + 2*sin (x)/*cos(2*x)
------------------------------------------
                          2               
           -cos(x) + 2*sin (x)
(2sin⁡2(x)−cos⁡(x))cos⁡(2x)−sin⁡2(x)2sin⁡2(x)−cos⁡(x)\frac{\left(2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(2 x \right)} - \sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}} 
(-sin(x)^2 + (-cos(x) + 2*sin(x)^2)*cos(2*x))/(-cos(x) + 2*sin(x)^2)
Potencias [src] 
                                             2           
 -2*I*x    2*I*x             /   -I*x    I*x\            
e         e                  \- e     + e   /            
------- + ------ + --------------------------------------
   2        2        /                  2               \
                     |  /   -I*x    I*x\     I*x    -I*x|
                     |  \- e     + e   /    e      e    |
                   4*|- ----------------- - ---- - -----|
                     \          2            2       2  /
(eix−e−ix)24(−(eix−e−ix)22−eix2−e−ix2)+e2ix2+e−2ix2\frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4 \left(- \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{2} - \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}\right)} + \frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2} 
exp(-2*i*x)/2 + exp(2*i*x)/2 + (-exp(-i*x) + exp(i*x))^2/(4*(-(-exp(-i*x) + exp(i*x))^2/2 - exp(i*x)/2 - exp(-i*x)/2))
Combinatoria [src] 
   2                             2            
sin (x) + cos(x)*cos(2*x) - 2*sin (x)*cos(2*x)
----------------------------------------------
                    2                         
             - 2*sin (x) + cos(x)
−2sin⁡2(x)cos⁡(2x)+sin⁡2(x)+cos⁡(x)cos⁡(2x)−2sin⁡2(x)+cos⁡(x)\frac{- 2 \sin^{2}{\left(x \right)} \cos{\left(2 x \right)} + \sin^{2}{\left(x \right)} + \cos{\left(x \right)} \cos{\left(2 x \right)}}{- 2 \sin^{2}{\left(x \right)} + \cos{\left(x \right)}} 
(sin(x)^2 + cos(x)*cos(2*x) - 2*sin(x)^2*cos(2*x))/(-2*sin(x)^2 + cos(x))
Unión de expresiones racionales [src] 
     2      /               2   \         
- sin (x) + \-cos(x) + 2*sin (x)/*cos(2*x)
------------------------------------------
                          2               
           -cos(x) + 2*sin (x)
(2sin⁡2(x)−cos⁡(x))cos⁡(2x)−sin⁡2(x)2sin⁡2(x)−cos⁡(x)\frac{\left(2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(2 x \right)} - \sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}} 
(-sin(x)^2 + (-cos(x) + 2*sin(x)^2)*cos(2*x))/(-cos(x) + 2*sin(x)^2)
Abrimos la expresión [src] 
                         2         
          2           sin (x)      
-1 + 2*cos (x) - ------------------
                      2            
                 2*sin (x) - cos(x)
2cos⁡2(x)−1−sin⁡2(x)2sin⁡2(x)−cos⁡(x)2 \cos^{2}{\left(x \right)} - 1 - \frac{\sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)}} 
-1 + 2*cos(x)^2 - sin(x)^2/(2*sin(x)^2 - cos(x))
Denominador común [src] 
         2                     
      sin (x)                  
-------------------- + cos(2*x)
       2                       
- 2*sin (x) + cos(x)
cos⁡(2x)+sin⁡2(x)−2sin⁡2(x)+cos⁡(x)\cos{\left(2 x \right)} + \frac{\sin^{2}{\left(x \right)}}{- 2 \sin^{2}{\left(x \right)} + \cos{\left(x \right)}} 
sin(x)^2/(-2*sin(x)^2 + cos(x)) + cos(2*x)
Parte trigonométrica [src] 
        -1 + cos(2*x)                  
- -------------------------- + cos(2*x)
  2*(-1 + cos(x) + cos(2*x))
−cos⁡(2x)−12(cos⁡(x)+cos⁡(2x)−1)+cos⁡(2x)- \frac{\cos{\left(2 x \right)} - 1}{2 \left(\cos{\left(x \right)} + \cos{\left(2 x \right)} - 1\right)} + \cos{\left(2 x \right)} 
           2/    pi\                 
        cos |x - --|                 
            \    2 /                 
- ------------------------ + cos(2*x)
                 2/    pi\           
  -cos(x) + 2*cos |x - --|           
                  \    2 /
cos⁡(2x)−cos⁡2(x−π2)−cos⁡(x)+2cos⁡2(x−π2)\cos{\left(2 x \right)} - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{- \cos{\left(x \right)} + 2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}} 
                   /pi      \                         
           -1 + sin|-- + 2*x|                         
                   \2       /               /pi      \
- ------------------------------------ + sin|-- + 2*x|
    /        /    pi\      /pi      \\      \2       /
  2*|-1 + sin|x + --| + sin|-- + 2*x||                
    \        \    2 /      \2       //
−sin⁡(2x+π2)−12(sin⁡(x+π2)+sin⁡(2x+π2)−1)+sin⁡(2x+π2)- \frac{\sin{\left(2 x + \frac{\pi}{2} \right)} - 1}{2 \left(\sin{\left(x + \frac{\pi}{2} \right)} + \sin{\left(2 x + \frac{\pi}{2} \right)} - 1\right)} + \sin{\left(2 x + \frac{\pi}{2} \right)} 
   1                    1              
-------- - ----------------------------
sec(2*x)   /    1         2   \    2   
           |- ------ + -------|*csc (x)
           |  sec(x)      2   |        
           \           csc (x)/
1sec⁡(2x)−1(−1sec⁡(x)+2csc⁡2(x))csc⁡2(x)\frac{1}{\sec{\left(2 x \right)}} - \frac{1}{\left(- \frac{1}{\sec{\left(x \right)}} + \frac{2}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}} 
                                   2            
                            1 - tan (x)         
                       -1 + -----------         
       2                           2            
1 - tan (x)                 1 + tan (x)         
----------- - ----------------------------------
       2        /                          2/x\\
1 + tan (x)     |            2      1 - tan |-||
                |     1 - tan (x)           \2/|
              2*|-1 + ----------- + -----------|
                |            2             2/x\|
                |     1 + tan (x)   1 + tan |-||
                \                           \2//
1−tan⁡2(x)tan⁡2(x)+1−1−tan⁡2(x)tan⁡2(x)+1−12(1−tan⁡2(x2)tan⁡2(x2)+1+1−tan⁡2(x)tan⁡2(x)+1−1)\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - 1}{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - 1\right)} 
                                       2/x\                    
        2                         4*cot |-|                    
-1 + cot (x)                            \2/                    
------------ - ------------------------------------------------
       2                      /          2/x\          2/x\   \
1 + cot (x)                 2 |  -1 + cot |-|     8*cot |-|   |
               /       2/x\\  |           \2/           \2/   |
               |1 + cot |-|| *|- ------------ + --------------|
               \        \2//  |         2/x\                 2|
                              |  1 + cot |-|    /       2/x\\ |
                              |          \2/    |1 + cot |-|| |
                              \                 \        \2// /
cot⁡2(x)−1cot⁡2(x)+1−4cot⁡2(x2)(−cot⁡2(x2)−1cot⁡2(x2)+1+8cot⁡2(x2)(cot⁡2(x2)+1)2)(cot⁡2(x2)+1)2\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - \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} + \frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
                                     2             
                             -1 + cot (x)          
                        -1 + ------------          
        2                           2              
-1 + cot (x)                 1 + cot (x)           
------------ - ------------------------------------
       2         /                            2/x\\
1 + cot (x)      |             2      -1 + cot |-||
                 |     -1 + cot (x)            \2/|
               2*|-1 + ------------ + ------------|
                 |            2              2/x\ |
                 |     1 + cot (x)    1 + cot |-| |
                 \                            \2/ /
−cot⁡2(x)−1cot⁡2(x)+1−12(cot⁡2(x2)−1cot⁡2(x2)+1+cot⁡2(x)−1cot⁡2(x)+1−1)+cot⁡2(x)−1cot⁡2(x)+1- \frac{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - 1}{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - 1\right)} + \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} 
                                      2/x\                   
       2                         4*tan |-|                   
1 - tan (x)                            \2/                   
----------- - -----------------------------------------------
       2                     /         2/x\          2/x\   \
1 + tan (x)                2 |  1 - tan |-|     8*tan |-|   |
              /       2/x\\  |          \2/           \2/   |
              |1 + tan |-|| *|- ----------- + --------------|
              \        \2//  |         2/x\                2|
                             |  1 + tan |-|   /       2/x\\ |
                             |          \2/   |1 + tan |-|| |
                             \                \        \2// /
1−tan⁡2(x)tan⁡2(x)+1−4tan⁡2(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+8tan⁡2(x2)(tan⁡2(x2)+1)2)(tan⁡2(x2)+1)2\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \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} + \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} 
              2                            
           sin (x)               /pi      \
- ------------------------- + sin|-- + 2*x|
       /    pi\        2         \2       /
  - sin|x + --| + 2*sin (x)                
       \    2 /
sin⁡(2x+π2)−sin⁡2(x)2sin⁡2(x)−sin⁡(x+π2)\sin{\left(2 x + \frac{\pi}{2} \right)} - \frac{\sin^{2}{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - \sin{\left(x + \frac{\pi}{2} \right)}} 
      1                         1                
------------- - ---------------------------------
   /pi      \   /       1           2   \    2   
csc|-- - 2*x|   |- ----------- + -------|*csc (x)
   \2       /   |     /pi    \      2   |        
                |  csc|-- - x|   csc (x)|        
                \     \2     /          /
1csc⁡(−2x+π2)−1(−1csc⁡(−x+π2)+2csc⁡2(x))csc⁡2(x)\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{\left(- \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}} 
                         1           
                 -1 + --------       
   1                  sec(2*x)       
-------- - --------------------------
sec(2*x)     /       1         1    \
           2*|-1 + ------ + --------|
             \     sec(x)   sec(2*x)/
−−1+1sec⁡(2x)2(−1+1sec⁡(2x)+1sec⁡(x))+1sec⁡(2x)- \frac{-1 + \frac{1}{\sec{\left(2 x \right)}}}{2 \left(-1 + \frac{1}{\sec{\left(2 x \right)}} + \frac{1}{\sec{\left(x \right)}}\right)} + \frac{1}{\sec{\left(2 x \right)}} 
   1                         1                   
-------- - --------------------------------------
sec(2*x)   /    1           2      \    2/    pi\
           |- ------ + ------------|*sec |x - --|
           |  sec(x)      2/    pi\|     \    2 /
           |           sec |x - --||             
           \               \    2 //
1sec⁡(2x)−1(2sec⁡2(x−π2)−1sec⁡(x))sec⁡2(x−π2)\frac{1}{\sec{\left(2 x \right)}} - \frac{1}{\left(\frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)}} 
                                    1               
                         -1 + -------------         
                                 /pi      \         
                              csc|-- - 2*x|         
      1                          \2       /         
------------- - ------------------------------------
   /pi      \     /          1              1      \
csc|-- - 2*x|   2*|-1 + ----------- + -------------|
   \2       /     |        /pi    \      /pi      \|
                  |     csc|-- - x|   csc|-- - 2*x||
                  \        \2     /      \2       //
−−1+1csc⁡(−2x+π2)2(−1+1csc⁡(−x+π2)+1csc⁡(−2x+π2))+1csc⁡(−2x+π2)- \frac{-1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}}{2 \left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)} + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} 
1/csc(pi/2 - 2*x) - (-1 + 1/csc(pi/2 - 2*x))/(2*(-1 + 1/csc(pi/2 - x) + 1/csc(pi/2 - 2*x)))
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