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Simplificación de expresiones/ Descomposición en fracciones simples/ (cos(2*a)-cos(4*a))/(cos(2*a)+cos(4*a))

¿Cómo vas a descomponer esta (cos(2*a)-cos(4*a))/(cos(2*a)+cos(4*a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(2*a) - cos(4*a)
-------------------
cos(2*a) + cos(4*a)
$$\frac{\cos{\left(2 a \right)} - \cos{\left(4 a \right)}}{\cos{\left(2 a \right)} + \cos{\left(4 a \right)}}$$ 
(cos(2*a) - cos(4*a))/(cos(2*a) + cos(4*a))
Simplificación general [src] 
   2           2   
tan (a) - 4*sin (a)
-------------------
             2     
   -1 + 4*sin (a)
$$\frac{- 4 \sin^{2}{\left(a \right)} + \tan^{2}{\left(a \right)}}{4 \sin^{2}{\left(a \right)} - 1}$$ 
(tan(a)^2 - 4*sin(a)^2)/(-1 + 4*sin(a)^2)
Respuesta numérica [src] 
(-cos(4*a) + cos(2*a))/(cos(2*a) + cos(4*a))
(-cos(4*a) + cos(2*a))/(cos(2*a) + cos(4*a))
Potencias [src] 
 -2*I*a    2*I*a    -4*I*a    4*I*a
e         e        e         e     
------- + ------ - ------- - ------
   2        2         2        2   
-----------------------------------
 -4*I*a    -2*I*a    2*I*a    4*I*a
e         e         e        e     
------- + ------- + ------ + ------
   2         2        2        2
$$\frac{- \frac{e^{4 i a}}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2} - \frac{e^{- 4 i a}}{2}}{\frac{e^{4 i a}}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2} + \frac{e^{- 4 i a}}{2}}$$ 
(exp(-2*i*a)/2 + exp(2*i*a)/2 - exp(-4*i*a)/2 - exp(4*i*a)/2)/(exp(-4*i*a)/2 + exp(-2*i*a)/2 + exp(2*i*a)/2 + exp(4*i*a)/2)
Denominador común [src] 
         2*cos(4*a)    
1 - -------------------
    cos(2*a) + cos(4*a)
$$1 - \frac{2 \cos{\left(4 a \right)}}{\cos{\left(2 a \right)} + \cos{\left(4 a \right)}}$$ 
1 - 2*cos(4*a)/(cos(2*a) + cos(4*a))
Abrimos la expresión [src] 
                                        4                          2         
             2                     8*cos (a)                 10*cos (a)      
- ----------------------- - ----------------------- + -----------------------
         2           4             2           4             2           4   
  - 6*cos (a) + 8*cos (a)   - 6*cos (a) + 8*cos (a)   - 6*cos (a) + 8*cos (a)
$$- \frac{8 \cos^{4}{\left(a \right)}}{8 \cos^{4}{\left(a \right)} - 6 \cos^{2}{\left(a \right)}} + \frac{10 \cos^{2}{\left(a \right)}}{8 \cos^{4}{\left(a \right)} - 6 \cos^{2}{\left(a \right)}} - \frac{2}{8 \cos^{4}{\left(a \right)} - 6 \cos^{2}{\left(a \right)}}$$ 
-2/(-6*cos(a)^2 + 8*cos(a)^4) - 8*cos(a)^4/(-6*cos(a)^2 + 8*cos(a)^4) + 10*cos(a)^2/(-6*cos(a)^2 + 8*cos(a)^4)
Parte trigonométrica [src] 
                      2      
       4           sec (a)   
- ------------ + ------------
     2/    pi\      2/    pi\
  sec |a - --|   sec |a - --|
      \    2 /       \    2 /
-----------------------------
                4            
      -1 + ------------      
              2/    pi\      
           sec |a - --|      
               \    2 /
$$\frac{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{4}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}}{-1 + \frac{4}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}}$$ 
                  2/a\  
            16*tan |-|  
   2               \2/  
tan (a) - --------------
                       2
          /       2/a\\ 
          |1 + tan |-|| 
          \        \2// 
------------------------
               2/a\     
         16*tan |-|     
                \2/     
  -1 + --------------   
                    2   
       /       2/a\\    
       |1 + tan |-||    
       \        \2//
$$\frac{\tan^{2}{\left(a \right)} - \frac{16 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{-1 + \frac{16 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}$$ 
   1          1    
-------- - --------
sec(2*a)   sec(4*a)
-------------------
   1          1    
-------- + --------
sec(2*a)   sec(4*a)
$$\frac{- \frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\sec{\left(2 a \right)}}}{\frac{1}{\sec{\left(4 a \right)}} + \frac{1}{\sec{\left(2 a \right)}}}$$ 
      1               1      
------------- - -------------
   /pi      \      /pi      \
csc|-- - 2*a|   csc|-- - 4*a|
   \2       /      \2       /
-----------------------------
      1               1      
------------- + -------------
   /pi      \      /pi      \
csc|-- - 4*a|   csc|-- - 2*a|
   \2       /      \2       /
$$\frac{\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}}{\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}}$$ 
                  2/a\  
            16*cot |-|  
   1               \2/  
------- - --------------
   2                   2
cot (a)   /       2/a\\ 
          |1 + cot |-|| 
          \        \2// 
------------------------
               2/a\     
         16*cot |-|     
                \2/     
  -1 + --------------   
                    2   
       /       2/a\\    
       |1 + cot |-||    
       \        \2//
$$\frac{\frac{1}{\cot^{2}{\left(a \right)}} - \frac{16 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{-1 + \frac{16 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}$$ 
                      2/    pi\
                   cos |a - --|
       2/    pi\       \    2 /
- 4*cos |a - --| + ------------
        \    2 /        2      
                     cos (a)   
-------------------------------
                2/    pi\      
      -1 + 4*cos |a - --|      
                 \    2 /
$$\frac{- 4 \cos^{2}{\left(a - \frac{\pi}{2} \right)} + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{4 \cos^{2}{\left(a - \frac{\pi}{2} \right)} - 1}$$ 
               2/pi    \
            csc |-- - a|
     4          \2     /
- ------- + ------------
     2           2      
  csc (a)     csc (a)   
------------------------
              4         
      -1 + -------      
              2         
           csc (a)
$$\frac{\frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}} - \frac{4}{\csc^{2}{\left(a \right)}}}{-1 + \frac{4}{\csc^{2}{\left(a \right)}}}$$ 
     /pi      \      /pi      \
- sin|-- + 4*a| + sin|-- + 2*a|
     \2       /      \2       /
-------------------------------
    /pi      \      /pi      \ 
 sin|-- + 2*a| + sin|-- + 4*a| 
    \2       /      \2       /
$$\frac{\sin{\left(2 a + \frac{\pi}{2} \right)} - \sin{\left(4 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a + \frac{\pi}{2} \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)}}$$ 
   2           2   
tan (a) - 4*sin (a)
-------------------
             2     
   -1 + 4*sin (a)
$$\frac{- 4 \sin^{2}{\left(a \right)} + \tan^{2}{\left(a \right)}}{4 \sin^{2}{\left(a \right)} - 1}$$ 
       2             2     
1 - tan (a)   1 - tan (2*a)
----------- - -------------
       2             2     
1 + tan (a)   1 + tan (2*a)
---------------------------
       2             2     
1 - tan (a)   1 - tan (2*a)
----------- + -------------
       2             2     
1 + tan (a)   1 + tan (2*a)
$$\frac{\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} - \frac{1 - \tan^{2}{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}}{\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \frac{1 - \tan^{2}{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}}$$ 
        2              2     
-1 + cot (a)   -1 + cot (2*a)
------------ - --------------
       2              2      
1 + cot (a)    1 + cot (2*a) 
-----------------------------
        2              2     
-1 + cot (a)   -1 + cot (2*a)
------------ + --------------
       2              2      
1 + cot (a)    1 + cot (2*a)
$$\frac{\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} - \frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1}}{\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + \frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1}}$$ 
                   4   
       2      4*sin (a)
- 4*sin (a) + ---------
                 2     
              sin (2*a)
-----------------------
               2       
     -1 + 4*sin (a)
$$\frac{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} - 4 \sin^{2}{\left(a \right)}}{4 \sin^{2}{\left(a \right)} - 1}$$ 
(-4*sin(a)^2 + 4*sin(a)^4/sin(2*a)^2)/(-1 + 4*sin(a)^2)
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