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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
      cos(4*a)     
-------------------
cos(2*a) - sin(2*a)
$$\frac{\cos{\left(4 a \right)}}{- \sin{\left(2 a \right)} + \cos{\left(2 a \right)}}$$ 
cos(4*a)/(cos(2*a) - sin(2*a))
Simplificación general [src] 
   ___         
 \/ 2 *cos(4*a)
---------------
     /      pi\
2*cos|2*a + --|
     \      4 /
$$\frac{\sqrt{2} \cos{\left(4 a \right)}}{2 \cos{\left(2 a + \frac{\pi}{4} \right)}}$$ 
sqrt(2)*cos(4*a)/(2*cos(2*a + pi/4))
Respuesta numérica [src] 
cos(4*a)/(-sin(2*a) + cos(2*a))
cos(4*a)/(-sin(2*a) + cos(2*a))
Potencias [src] 
              -4*I*a    4*I*a            
             e         e                 
             ------- + ------            
                2        2               
-----------------------------------------
 -2*I*a    2*I*a     /   -2*I*a    2*I*a\
e         e        I*\- e       + e     /
------- + ------ + ----------------------
   2        2                2
$$\frac{\frac{e^{4 i a}}{2} + \frac{e^{- 4 i a}}{2}}{\frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}}$$ 
(exp(-4*i*a)/2 + exp(4*i*a)/2)/(exp(-2*i*a)/2 + exp(2*i*a)/2 + i*(-exp(-2*i*a) + exp(2*i*a))/2)
Abrimos la expresión [src] 
                                                   2                                  4               
               1                              8*cos (a)                          8*cos (a)            
-------------------------------- - -------------------------------- + --------------------------------
          2                                  2                                  2                     
-1 + 2*cos (a) - 2*cos(a)*sin(a)   -1 + 2*cos (a) - 2*cos(a)*sin(a)   -1 + 2*cos (a) - 2*cos(a)*sin(a)
$$\frac{8 \cos^{4}{\left(a \right)}}{- 2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1} - \frac{8 \cos^{2}{\left(a \right)}}{- 2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1} + \frac{1}{- 2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1}$$ 
1/(-1 + 2*cos(a)^2 - 2*cos(a)*sin(a)) - 8*cos(a)^2/(-1 + 2*cos(a)^2 - 2*cos(a)*sin(a)) + 8*cos(a)^4/(-1 + 2*cos(a)^2 - 2*cos(a)*sin(a))
Parte trigonométrica [src] 
  ___    /pi      \
\/ 2 *sin|-- + 4*a|
         \2       /
-------------------
       /      pi\  
  2*cos|2*a + --|  
       \      4 /
$$\frac{\sqrt{2} \sin{\left(4 a + \frac{\pi}{2} \right)}}{2 \cos{\left(2 a + \frac{\pi}{4} \right)}}$$ 
  ___ /       2/    pi\\ /        2     \
\/ 2 *|1 + cot |a + --||*\-1 + cot (2*a)/
      \        \    8 //                 
-----------------------------------------
    /       2     \ /        2/    pi\\  
  2*\1 + cot (2*a)/*|-1 + cot |a + --||  
                    \         \    8 //
$$\frac{\sqrt{2} \left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(2 a \right)} + 1\right) \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right)}$$ 
              1               
------------------------------
/   1          1    \         
|-------- - --------|*sec(4*a)
\sec(2*a)   csc(2*a)/
$$\frac{1}{\left(\frac{1}{\sec{\left(2 a \right)}} - \frac{1}{\csc{\left(2 a \right)}}\right) \sec{\left(4 a \right)}}$$ 
  ___    2      /       2/    pi\\ /       2     \
\/ 2 *cos (2*a)*|1 + tan |a + --||*\1 - tan (2*a)/
                \        \    8 //                
--------------------------------------------------
                 /       2/    pi\\               
               2*|1 - tan |a + --||               
                 \        \    8 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(2 a \right)}\right) \left(\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(2 a \right)}}{2 \left(1 - \tan^{2}{\left(a + \frac{\pi}{8} \right)}\right)}$$ 
  ___    /      pi\
\/ 2 *sec|2*a + --|
         \      4 /
-------------------
     2*sec(4*a)
$$\frac{\sqrt{2} \sec{\left(2 a + \frac{\pi}{4} \right)}}{2 \sec{\left(4 a \right)}}$$ 
         /pi      \      
      sin|-- + 4*a|      
         \2       /      
-------------------------
               /pi      \
-sin(2*a) + sin|-- + 2*a|
               \2       /
$$\frac{\sin{\left(4 a + \frac{\pi}{2} \right)}}{- \sin{\left(2 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}}$$ 
  ___    /       pi\
\/ 2 *csc|-2*a + --|
         \       4 /
--------------------
     2*sec(4*a)
$$\frac{\sqrt{2} \csc{\left(- 2 a + \frac{\pi}{4} \right)}}{2 \sec{\left(4 a \right)}}$$ 
      ___    
    \/ 2     
-------------
   /      pi\
csc|2*a + --|
   \      4 /
$$\frac{\sqrt{2}}{\csc{\left(2 a + \frac{\pi}{4} \right)}}$$ 
  ___    /      pi\
\/ 2 *sin|2*a + --|
         \      4 /
$$\sqrt{2} \sin{\left(2 a + \frac{\pi}{4} \right)}$$ 
                        ___                              
                     -\/ 2 *cos(4*a)                     
---------------------------------------------------------
                      ___________    ___________         
    ___              /       ___    /       ___          
- \/ 2 *cos(2*a) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(2*a)
$$- \frac{\sqrt{2} \cos{\left(4 a \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(2 a \right)} - \sqrt{2} \cos{\left(2 a \right)}}$$ 
           ___          
         \/ 2           
------------------------
     /      pi\         
2*cos|2*a + --|*sec(4*a)
     \      4 /
$$\frac{\sqrt{2}}{2 \cos{\left(2 a + \frac{\pi}{4} \right)} \sec{\left(4 a \right)}}$$ 
  ___    /pi      \
\/ 2 *sin|-- + 4*a|
         \2       /
-------------------
      /      3*pi\ 
 2*sin|2*a + ----| 
      \       4  /
$$\frac{\sqrt{2} \sin{\left(4 a + \frac{\pi}{2} \right)}}{2 \sin{\left(2 a + \frac{3 \pi}{4} \right)}}$$ 
    ___          
  \/ 2 *cos(4*a) 
-----------------
     /      3*pi\
2*sin|2*a + ----|
     \       4  /
$$\frac{\sqrt{2} \cos{\left(4 a \right)}}{2 \sin{\left(2 a + \frac{3 \pi}{4} \right)}}$$ 
      ___    
    \/ 2     
-------------
   /      pi\
sec|2*a - --|
   \      4 /
$$\frac{\sqrt{2}}{\sec{\left(2 a - \frac{\pi}{4} \right)}}$$ 
  ___    /       pi\
\/ 2 *csc|-2*a + --|
         \       4 /
--------------------
       /pi      \   
  2*csc|-- - 4*a|   
       \2       /
$$\frac{\sqrt{2} \csc{\left(- 2 a + \frac{\pi}{4} \right)}}{2 \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$ 
                       2                    
               -1 + cot (2*a)               
--------------------------------------------
                /        2                 \
/       2     \ |-1 + cot (a)     2*cot(a) |
\1 + cot (2*a)/*|------------ - -----------|
                |       2              2   |
                \1 + cot (a)    1 + cot (a)/
$$\frac{\cot^{2}{\left(2 a \right)} - 1}{\left(\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} - \frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}\right) \left(\cot^{2}{\left(2 a \right)} + 1\right)}$$ 
                   ___    /      pi\                 
                 \/ 2 *tan|2*a + --|                 
                          \      4 /                 
-----------------------------------------------------
/       2/      pi\\ /        2/    pi\\    2/    pi\
|1 + tan |2*a + --||*|-1 + cot |a + --||*sin |a + --|
\        \      4 // \         \    8 //     \    8 /
$$\frac{\sqrt{2} \tan{\left(2 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(a + \frac{\pi}{8} \right)}}$$ 
   ___         
 \/ 2 *cos(4*a)
---------------
     /      pi\
2*cos|2*a + --|
     \      4 /
$$\frac{\sqrt{2} \cos{\left(4 a \right)}}{2 \cos{\left(2 a + \frac{\pi}{4} \right)}}$$ 
              ___            
            \/ 2             
-----------------------------
     /      pi\    /pi      \
2*cos|2*a + --|*csc|-- - 4*a|
     \      4 /    \2       /
$$\frac{\sqrt{2}}{2 \cos{\left(2 a + \frac{\pi}{4} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$ 
   ___ /       2/    pi\\    /      pi\ 
 \/ 2 *|1 + cot |a + --||*tan|2*a + --| 
       \        \    8 //    \      4 / 
----------------------------------------
/       2/      pi\\ /        2/    pi\\
|1 + tan |2*a + --||*|-1 + cot |a + --||
\        \      4 // \         \    8 //
$$\frac{\sqrt{2} \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right)}$$ 
  ___    /      pi\
\/ 2 *cos|2*a - --|
         \      4 /
$$\sqrt{2} \cos{\left(2 a - \frac{\pi}{4} \right)}$$ 
                 1                 
-----------------------------------
/   1             1      \         
|-------- - -------------|*sec(4*a)
|sec(2*a)      /      pi\|         
|           sec|2*a - --||         
\              \      2 //
$$\frac{1}{\left(- \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}\right) \sec{\left(4 a \right)}}$$ 
   ___ /       2/    pi\\    /      pi\
 \/ 2 *|1 + tan |a + --||*tan|2*a + --|
       \        \    8 //    \      4 /
---------------------------------------
/       2/      pi\\ /       2/    pi\\
|1 + tan |2*a + --||*|1 - tan |a + --||
\        \      4 // \        \    8 //
$$\frac{\sqrt{2} \left(\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\left(1 - \tan^{2}{\left(a + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)}$$ 
  ___ /       2/    pi\\ /       2     \
\/ 2 *|1 + tan |a + --||*\1 - tan (2*a)/
      \        \    8 //                
----------------------------------------
    /       2     \ /       2/    pi\\  
  2*\1 + tan (2*a)/*|1 - tan |a + --||  
                    \        \    8 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(2 a \right)}\right) \left(\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(a + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(2 a \right)} + 1\right)}$$ 
                   1                    
----------------------------------------
/      1            1    \    /pi      \
|------------- - --------|*csc|-- - 4*a|
|   /pi      \   csc(2*a)|    \2       /
|csc|-- - 2*a|           |              
\   \2       /           /
$$\frac{1}{\left(\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(2 a \right)}}\right) \csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$ 
         cos(4*a)         
--------------------------
     /      pi\           
- cos|2*a - --| + cos(2*a)
     \      2 /
$$\frac{\cos{\left(4 a \right)}}{\cos{\left(2 a \right)} - \cos{\left(2 a - \frac{\pi}{2} \right)}}$$ 
    ___    2/    pi\    /    pi\
2*\/ 2 *sin |a + --|*cot|a + --|
            \    8 /    \    8 /
$$2 \sqrt{2} \sin^{2}{\left(a + \frac{\pi}{8} \right)} \cot{\left(a + \frac{\pi}{8} \right)}$$ 
   ___ /       2/    pi\\    /      pi\ 
 \/ 2 *|1 + cot |a + --||*cot|2*a + --| 
       \        \    8 //    \      4 / 
----------------------------------------
/       2/      pi\\ /        2/    pi\\
|1 + cot |2*a + --||*|-1 + cot |a + --||
\        \      4 // \         \    8 //
$$\frac{\sqrt{2} \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \cot{\left(2 a + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right) \left(\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)}$$ 
    ___    /    pi\
2*\/ 2 *tan|a + --|
           \    8 /
-------------------
         2/    pi\ 
  1 + tan |a + --| 
          \    8 /
$$\frac{2 \sqrt{2} \tan{\left(a + \frac{\pi}{8} \right)}}{\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1}$$ 
                      2                    
               1 - tan (2*a)               
-------------------------------------------
                /       2                 \
/       2     \ |1 - tan (a)     2*tan(a) |
\1 + tan (2*a)/*|----------- - -----------|
                |       2             2   |
                \1 + tan (a)   1 + tan (a)/
$$\frac{1 - \tan^{2}{\left(2 a \right)}}{\left(\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} - \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}\right) \left(\tan^{2}{\left(2 a \right)} + 1\right)}$$ 
    ___    /    pi\
2*\/ 2 *cot|a + --|
           \    8 /
-------------------
         2/    pi\ 
  1 + cot |a + --| 
          \    8 /
$$\frac{2 \sqrt{2} \cot{\left(a + \frac{\pi}{8} \right)}}{\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1}$$ 
2*sqrt(2)*cot(a + pi/8)/(1 + cot(a + pi/8)^2)
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