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Simplificación de expresiones/ Descomposición en fracciones simples/ cos2x-sin2x/(cos4x-sin4x+2sin^2(2x))

¿Cómo vas a descomponer esta cos2x-sin2x/(cos4x-sin4x+2sin^2(2x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                        sin(2*x)            
cos(2*x) - ---------------------------------
                                      2     
           cos(4*x) - sin(4*x) + 2*sin (2*x)
cos⁡(2x)−sin⁡(2x)(−sin⁡(4x)+cos⁡(4x))+2sin⁡2(2x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) + 2 \sin^{2}{\left(2 x \right)}} 
cos(2*x) - sin(2*x)/(cos(4*x) - sin(4*x) + 2*sin(2*x)^2)
Simplificación general [src] 
-sin(2*x) + (1 - sin(4*x))*cos(2*x)
-----------------------------------
            1 - sin(4*x)
(1−sin⁡(4x))cos⁡(2x)−sin⁡(2x)1−sin⁡(4x)\frac{\left(1 - \sin{\left(4 x \right)}\right) \cos{\left(2 x \right)} - \sin{\left(2 x \right)}}{1 - \sin{\left(4 x \right)}} 
(-sin(2*x) + (1 - sin(4*x))*cos(2*x))/(1 - sin(4*x))
Denominador común [src] 
               sin(2*x)                        
- ---------------------------------- + cos(2*x)
                   2                           
  -sin(4*x) + 2*sin (2*x) + cos(4*x)
cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
-sin(2*x)/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x)) + cos(2*x)
Respuesta numérica [src] 
-sin(2*x)/(-sin(4*x) + 2.0*sin(2*x)^2 + cos(4*x)) + cos(2*x)
-sin(2*x)/(-sin(4*x) + 2.0*sin(2*x)^2 + cos(4*x)) + cos(2*x)
Potencias [src] 
 -2*I*x    2*I*x                             /   -2*I*x    2*I*x\                       
e         e                                I*\- e       + e     /                       
------- + ------ + ---------------------------------------------------------------------
   2        2        /                                       2                         \
                     | -4*I*x    4*I*x   /   -2*I*x    2*I*x\      /   -4*I*x    4*I*x\|
                     |e         e        \- e       + e     /    I*\- e       + e     /|
                   2*|------- + ------ - --------------------- + ----------------------|
                     \   2        2                2                       2           /
i(e2ix−e−2ix)2(−(e2ix−e−2ix)22+i(e4ix−e−4ix)2+e4ix2+e−4ix2)+e2ix2+e−2ix2\frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{2 \left(- \frac{\left(e^{2 i x} - e^{- 2 i x}\right)^{2}}{2} + \frac{i \left(e^{4 i x} - e^{- 4 i x}\right)}{2} + \frac{e^{4 i x}}{2} + \frac{e^{- 4 i x}}{2}\right)} + \frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2} 
               sin(2*x)                        
- ---------------------------------- + cos(2*x)
                   2                           
  -sin(4*x) + 2*sin (2*x) + cos(4*x)
cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
-sin(2*x)/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x)) + cos(2*x)
Compilar la expresión [src] 
               sin(2*x)                        
- ---------------------------------- + cos(2*x)
                   2                           
  -sin(4*x) + 2*sin (2*x) + cos(4*x)
cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
-sin(2*x)/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x)) + cos(2*x)
Unión de expresiones racionales [src] 
            /                 2                \         
-sin(2*x) + \-sin(4*x) + 2*sin (2*x) + cos(4*x)/*cos(2*x)
---------------------------------------------------------
                             2                           
            -sin(4*x) + 2*sin (2*x) + cos(4*x)
(2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x))cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\frac{\left(2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) \cos{\left(2 x \right)} - \sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
(-sin(2*x) + (-sin(4*x) + 2*sin(2*x)^2 + cos(4*x))*cos(2*x))/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x))
Denominador racional [src] 
            /                 2                \         
-sin(2*x) + \-sin(4*x) + 2*sin (2*x) + cos(4*x)/*cos(2*x)
---------------------------------------------------------
                             2                           
            -sin(4*x) + 2*sin (2*x) + cos(4*x)
(2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x))cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\frac{\left(2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) \cos{\left(2 x \right)} - \sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
(-sin(2*x) + (-sin(4*x) + 2*sin(2*x)^2 + cos(4*x))*cos(2*x))/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x))
Abrimos la expresión [src] 
          2                                       2*cos(x)*sin(x)                                  
-1 + 2*cos (x) - ----------------------------------------------------------------------------------
                          2           4                             2       2           3          
                 1 - 8*cos (x) + 8*cos (x) - 4*cos(x)*sin(x) + 8*cos (x)*sin (x) + 8*sin (x)*cos(x)
2cos⁡2(x)−1−2sin⁡(x)cos⁡(x)8sin⁡3(x)cos⁡(x)+8sin⁡2(x)cos⁡2(x)−4sin⁡(x)cos⁡(x)+8cos⁡4(x)−8cos⁡2(x)+12 \cos^{2}{\left(x \right)} - 1 - \frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{8 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - 4 \sin{\left(x \right)} \cos{\left(x \right)} + 8 \cos^{4}{\left(x \right)} - 8 \cos^{2}{\left(x \right)} + 1} 
-1 + 2*cos(x)^2 - 2*cos(x)*sin(x)/(1 - 8*cos(x)^2 + 8*cos(x)^4 - 4*cos(x)*sin(x) + 8*cos(x)^2*sin(x)^2 + 8*sin(x)^3*cos(x))
Combinatoria [src] 
                                                         2              
-sin(2*x) + cos(2*x)*cos(4*x) - cos(2*x)*sin(4*x) + 2*sin (2*x)*cos(2*x)
------------------------------------------------------------------------
                                    2                                   
                   -sin(4*x) + 2*sin (2*x) + cos(4*x)
2sin⁡2(2x)cos⁡(2x)−sin⁡(2x)−sin⁡(4x)cos⁡(2x)+cos⁡(2x)cos⁡(4x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\frac{2 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} - \sin{\left(4 x \right)} \cos{\left(2 x \right)} + \cos{\left(2 x \right)} \cos{\left(4 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
(-sin(2*x) + cos(2*x)*cos(4*x) - cos(2*x)*sin(4*x) + 2*sin(2*x)^2*cos(2*x))/(-sin(4*x) + 2*sin(2*x)^2 + cos(4*x))
Parte trigonométrica [src] 
   1                                   1                            
-------- - ---------------------------------------------------------
sec(2*x)   /   1             1               2       \    /      pi\
           |-------- - ------------- + --------------|*sec|2*x - --|
           |sec(4*x)      /      pi\      2/      pi\|    \      2 /
           |           sec|4*x - --|   sec |2*x - --||              
           \              \      2 /       \      2 //
1sec⁡(2x)−1(−1sec⁡(4x−π2)+2sec⁡2(2x−π2)+1sec⁡(4x))sec⁡(2x−π2)\frac{1}{\sec{\left(2 x \right)}} - \frac{1}{\left(- \frac{1}{\sec{\left(4 x - \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 x \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}} 
        2                                                                     
-1 + cot (x)                               2*cot(x)                           
------------ - ---------------------------------------------------------------
       2                     /        2                               2      \
1 + cot (x)    /       2   \ |-1 + cot (2*x)     2*cot(2*x)      8*cot (x)   |
               \1 + cot (x)/*|-------------- - ------------- + --------------|
                             |       2                2                     2|
                             |1 + cot (2*x)    1 + cot (2*x)   /       2   \ |
                             \                                 \1 + cot (x)/ /
cot⁡2(x)−1cot⁡2(x)+1−2cot⁡(x)(cot⁡2(x)+1)(cot⁡2(2x)−1cot⁡2(2x)+1−2cot⁡(2x)cot⁡2(2x)+1+8cot⁡2(x)(cot⁡2(x)+1)2)\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - \frac{2 \cot{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} - \frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} + \frac{8 \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)} 
       2                                       
1 - tan (x)                2*tan(x)            
----------- - ---------------------------------
       2      /       2   \ /      2*tan(2*x) \
1 + tan (x)   \1 + tan (x)/*|1 - -------------|
                            |           2     |
                            \    1 + tan (2*x)/
1−tan⁡2(x)tan⁡2(x)+1−2tan⁡(x)(1−2tan⁡(2x)tan⁡2(2x)+1)(tan⁡2(x)+1)\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{2 \tan{\left(x \right)}}{\left(1 - \frac{2 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} 
      1                    1           
------------- - -----------------------
   /pi      \   /       1    \         
csc|-- - 2*x|   |1 - --------|*csc(2*x)
   \2       /   \    csc(4*x)/
1csc⁡(−2x+π2)−1(1−1csc⁡(4x))csc⁡(2x)\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(4 x \right)}}\right) \csc{\left(2 x \right)}} 
   1                       1                
-------- - ---------------------------------
sec(2*x)   /          1      \    /      pi\
           |1 - -------------|*sec|2*x - --|
           |       /      pi\|    \      2 /
           |    sec|4*x - --||              
           \       \      2 //
1sec⁡(2x)−1(1−1sec⁡(4x−π2))sec⁡(2x−π2)\frac{1}{\sec{\left(2 x \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(4 x - \frac{\pi}{2} \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}} 
               sin(2*x)                        
- ---------------------------------- + cos(2*x)
                   2                           
  -sin(4*x) + 2*sin (2*x) + cos(4*x)
cos⁡(2x)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+cos⁡(4x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \cos{\left(4 x \right)}} 
    sin(2*x)             
- ------------ + cos(2*x)
  1 - sin(4*x)
cos⁡(2x)−sin⁡(2x)1−sin⁡(4x)\cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{1 - \sin{\left(4 x \right)}} 
      1                                1                       
------------- - -----------------------------------------------
   /pi      \   /      1            1           2    \         
csc|-- - 2*x|   |------------- - -------- + ---------|*csc(2*x)
   \2       /   |   /pi      \   csc(4*x)      2     |         
                |csc|-- - 4*x|              csc (2*x)|         
                \   \2       /                       /
1csc⁡(−2x+π2)−1(1csc⁡(−4x+π2)−1csc⁡(4x)+2csc⁡2(2x))csc⁡(2x)\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{\left(\frac{1}{\csc{\left(- 4 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(4 x \right)}} + \frac{2}{\csc^{2}{\left(2 x \right)}}\right) \csc{\left(2 x \right)}} 
       2                                                                    
1 - tan (x)                              2*tan(x)                           
----------- - --------------------------------------------------------------
       2                    /       2                               2      \
1 + tan (x)   /       2   \ |1 - tan (2*x)     2*tan(2*x)      8*tan (x)   |
              \1 + tan (x)/*|------------- - ------------- + --------------|
                            |       2               2                     2|
                            |1 + tan (2*x)   1 + tan (2*x)   /       2   \ |
                            \                                \1 + tan (x)/ /
1−tan⁡2(x)tan⁡2(x)+1−2tan⁡(x)(tan⁡2(x)+1)(1−tan⁡2(2x)tan⁡2(2x)+1−2tan⁡(2x)tan⁡2(2x)+1+8tan⁡2(x)(tan⁡2(x)+1)2)\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{2 \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{1 - \tan^{2}{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1} - \frac{2 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1} + \frac{8 \tan^{2}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}\right)} 
                     /      pi\                           
                  cos|2*x - --|                           
                     \      2 /                           
- --------------------------------------------- + cos(2*x)
       /      pi\        2/      pi\                      
  - cos|4*x - --| + 2*cos |2*x - --| + cos(4*x)           
       \      2 /         \      2 /
cos⁡(2x)−cos⁡(2x−π2)cos⁡(4x)+2cos⁡2(2x−π2)−cos⁡(4x−π2)\cos{\left(2 x \right)} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(4 x \right)} + 2 \cos^{2}{\left(2 x - \frac{\pi}{2} \right)} - \cos{\left(4 x - \frac{\pi}{2} \right)}} 
       /      pi\             
    cos|2*x - --|             
       \      2 /             
- ----------------- + cos(2*x)
         /      pi\           
  1 - cos|4*x - --|           
         \      2 /
cos⁡(2x)−cos⁡(2x−π2)1−cos⁡(4x−π2)\cos{\left(2 x \right)} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{1 - \cos{\left(4 x - \frac{\pi}{2} \right)}} 
   1                           1                     
-------- - ------------------------------------------
sec(2*x)   /   1          1           2    \         
           |-------- - -------- + ---------|*csc(2*x)
           |sec(4*x)   csc(4*x)      2     |         
           \                      csc (2*x)/
1sec⁡(2x)−1(1sec⁡(4x)−1csc⁡(4x)+2csc⁡2(2x))csc⁡(2x)\frac{1}{\sec{\left(2 x \right)}} - \frac{1}{\left(\frac{1}{\sec{\left(4 x \right)}} - \frac{1}{\csc{\left(4 x \right)}} + \frac{2}{\csc^{2}{\left(2 x \right)}}\right) \csc{\left(2 x \right)}} 
        2                                       
-1 + cot (x)                2*cot(x)            
------------ - ---------------------------------
       2       /       2   \ /      2*cot(2*x) \
1 + cot (x)    \1 + cot (x)/*|1 - -------------|
                             |           2     |
                             \    1 + cot (2*x)/
cot⁡2(x)−1cot⁡2(x)+1−2cot⁡(x)(1−2cot⁡(2x)cot⁡2(2x)+1)(cot⁡2(x)+1)\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} - \frac{2 \cot{\left(x \right)}}{\left(1 - \frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)} 
                  sin(2*x)                     /pi      \
- --------------------------------------- + sin|-- + 2*x|
                   2           /pi      \      \2       /
  -sin(4*x) + 2*sin (2*x) + sin|-- + 4*x|                
                               \2       /
sin⁡(2x+π2)−sin⁡(2x)2sin⁡2(2x)−sin⁡(4x)+sin⁡(4x+π2)\sin{\left(2 x + \frac{\pi}{2} \right)} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(2 x \right)} - \sin{\left(4 x \right)} + \sin{\left(4 x + \frac{\pi}{2} \right)}} 
    sin(2*x)        /pi      \
- ------------ + sin|-- + 2*x|
  1 - sin(4*x)      \2       /
sin⁡(2x+π2)−sin⁡(2x)1−sin⁡(4x)\sin{\left(2 x + \frac{\pi}{2} \right)} - \frac{\sin{\left(2 x \right)}}{1 - \sin{\left(4 x \right)}} 
-sin(2*x)/(1 - sin(4*x)) + sin(pi/2 + 2*x)
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