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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(10*a)/(cos(5*a)-sin(5*a))

¿Cómo vas a descomponer esta cos(10*a)/(cos(5*a)-sin(5*a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
     cos(10*a)     
-------------------
cos(5*a) - sin(5*a)
$$\frac{\cos{\left(10 a \right)}}{- \sin{\left(5 a \right)} + \cos{\left(5 a \right)}}$$ 
cos(10*a)/(cos(5*a) - sin(5*a))
Simplificación general [src] 
  ___          
\/ 2 *cos(10*a)
---------------
     /      pi\
2*cos|5*a + --|
     \      4 /
$$\frac{\sqrt{2} \cos{\left(10 a \right)}}{2 \cos{\left(5 a + \frac{\pi}{4} \right)}}$$ 
sqrt(2)*cos(10*a)/(2*cos(5*a + pi/4))
Potencias [src] 
             -10*I*a    10*I*a           
            e          e                 
            -------- + -------           
               2          2              
-----------------------------------------
 -5*I*a    5*I*a     /   -5*I*a    5*I*a\
e         e        I*\- e       + e     /
------- + ------ + ----------------------
   2        2                2
$$\frac{\frac{e^{10 i a}}{2} + \frac{e^{- 10 i a}}{2}}{\frac{i \left(e^{5 i a} - e^{- 5 i a}\right)}{2} + \frac{e^{5 i a}}{2} + \frac{e^{- 5 i a}}{2}}$$ 
(exp(-10*i*a)/2 + exp(10*i*a)/2)/(exp(-5*i*a)/2 + exp(5*i*a)/2 + i*(-exp(-5*i*a) + exp(5*i*a))/2)
Abrimos la expresión [src] 
                                                                                                                     8                                                                          4                                                                           2                                                                           10                                                                           6                                 
                                      1                                                                      1280*cos (a)                                                                400*cos (a)                                                                  50*cos (a)                                                                 512*cos  (a)                                                                1120*cos (a)                              
- ------------------------------------------------------------------------- - ------------------------------------------------------------------------- - ------------------------------------------------------------------------- + ------------------------------------------------------------------------- + ------------------------------------------------------------------------- + -------------------------------------------------------------------------
          3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3              3            5                                  5            3   
  - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)   - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)   - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)   - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)   - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)   - 20*cos (a) - 16*sin (a) - 5*sin(a) + 5*cos(a) + 16*cos (a) + 20*sin (a)
$$\frac{512 \cos^{10}{\left(a \right)}}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{1280 \cos^{8}{\left(a \right)}}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} + \frac{1120 \cos^{6}{\left(a \right)}}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{400 \cos^{4}{\left(a \right)}}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} + \frac{50 \cos^{2}{\left(a \right)}}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{1}{- 16 \sin^{5}{\left(a \right)} + 20 \sin^{3}{\left(a \right)} - 5 \sin{\left(a \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}}$$ 
-1/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3) - 1280*cos(a)^8/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3) - 400*cos(a)^4/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3) + 50*cos(a)^2/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3) + 512*cos(a)^10/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3) + 1120*cos(a)^6/(-20*cos(a)^3 - 16*sin(a)^5 - 5*sin(a) + 5*cos(a) + 16*cos(a)^5 + 20*sin(a)^3)
Respuesta numérica [src] 
cos(10*a)/(-sin(5*a) + cos(5*a))
cos(10*a)/(-sin(5*a) + cos(5*a))
Parte trigonométrica [src] 
  ___    /       pi\
\/ 2 *csc|-5*a + --|
         \       4 /
--------------------
       /pi       \  
  2*csc|-- - 10*a|  
       \2        /
$$\frac{\sqrt{2} \csc{\left(- 5 a + \frac{\pi}{4} \right)}}{2 \csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$ 
  ___    2      /       2/pi   5*a\\ /       2     \
\/ 2 *cos (5*a)*|1 + tan |-- + ---||*\1 - tan (5*a)/
                \        \8     2 //                
----------------------------------------------------
                 /       2/pi   5*a\\               
               2*|1 - tan |-- + ---||               
                 \        \8     2 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 a \right)}\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(5 a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}\right)}$$ 
        cos(10*a)         
--------------------------
     /      pi\           
- cos|5*a - --| + cos(5*a)
     \      2 /
$$\frac{\cos{\left(10 a \right)}}{\cos{\left(5 a \right)} - \cos{\left(5 a - \frac{\pi}{2} \right)}}$$ 
                 1                  
------------------------------------
/   1             1      \          
|-------- - -------------|*sec(10*a)
|sec(5*a)      /      pi\|          
|           sec|5*a - --||          
\              \      2 //
$$\frac{1}{\left(- \frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
               1               
-------------------------------
/   1          1    \          
|-------- - --------|*sec(10*a)
\sec(5*a)   csc(5*a)/
$$\frac{1}{\left(\frac{1}{\sec{\left(5 a \right)}} - \frac{1}{\csc{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
                        2                      
                 1 - tan (5*a)                 
-----------------------------------------------
                /       2/5*a\          /5*a\ \
                |1 - tan |---|     2*tan|---| |
/       2     \ |        \ 2 /          \ 2 / |
\1 + tan (5*a)/*|------------- - -------------|
                |       2/5*a\          2/5*a\|
                |1 + tan |---|   1 + tan |---||
                \        \ 2 /           \ 2 //
$$\frac{1 - \tan^{2}{\left(5 a \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)}$$ 
    ___    /pi   5*a\
2*\/ 2 *cot|-- + ---|
           \8     2 /
---------------------
         2/pi   5*a\ 
  1 + cot |-- + ---| 
          \8     2 /
$$\frac{2 \sqrt{2} \cot{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1}$$ 
  ___    /      pi\
\/ 2 *cos|5*a - --|
         \      4 /
$$\sqrt{2} \cos{\left(5 a - \frac{\pi}{4} \right)}$$ 
  ___    /pi       \
\/ 2 *sin|-- + 10*a|
         \2        /
--------------------
      /      3*pi\  
 2*sin|5*a + ----|  
      \       4  /
$$\frac{\sqrt{2} \sin{\left(10 a + \frac{\pi}{2} \right)}}{2 \sin{\left(5 a + \frac{3 \pi}{4} \right)}}$$ 
  ___              /       pi\
\/ 2 *cos(10*a)*csc|-5*a + --|
                   \       4 /
------------------------------
              2
$$\frac{\sqrt{2} \cos{\left(10 a \right)} \csc{\left(- 5 a + \frac{\pi}{4} \right)}}{2}$$ 
                    1                    
-----------------------------------------
/      1            1    \    /pi       \
|------------- - --------|*csc|-- - 10*a|
|   /pi      \   csc(5*a)|    \2        /
|csc|-- - 5*a|           |               
\   \2       /           /
$$\frac{1}{\left(\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(5 a \right)}}\right) \csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$ 
      ___    
    \/ 2     
-------------
   /      pi\
sec|5*a - --|
   \      4 /
$$\frac{\sqrt{2}}{\sec{\left(5 a - \frac{\pi}{4} \right)}}$$ 
  ___ /       2/pi   5*a\\ /        2     \
\/ 2 *|1 + cot |-- + ---||*\-1 + cot (5*a)/
      \        \8     2 //                 
-------------------------------------------
    /       2     \ /        2/pi   5*a\\  
  2*\1 + cot (5*a)/*|-1 + cot |-- + ---||  
                    \         \8     2 //
$$\frac{\sqrt{2} \left(\cot^{2}{\left(5 a \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(5 a \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} - 1\right)}$$ 
  ___ /       2/pi   5*a\\ /       2     \
\/ 2 *|1 + tan |-- + ---||*\1 - tan (5*a)/
      \        \8     2 //                
------------------------------------------
    /       2     \ /       2/pi   5*a\\  
  2*\1 + tan (5*a)/*|1 - tan |-- + ---||  
                    \        \8     2 //
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 a \right)}\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)}$$ 
      ___    
    \/ 2     
-------------
   /      pi\
csc|5*a + --|
   \      4 /
$$\frac{\sqrt{2}}{\csc{\left(5 a + \frac{\pi}{4} \right)}}$$ 
  ___    /      pi\
\/ 2 *sin|5*a + --|
         \      4 /
$$\sqrt{2} \sin{\left(5 a + \frac{\pi}{4} \right)}$$ 
         /pi       \     
      sin|-- + 10*a|     
         \2        /     
-------------------------
               /pi      \
-sin(5*a) + sin|-- + 5*a|
               \2       /
$$\frac{\sin{\left(10 a + \frac{\pi}{2} \right)}}{- \sin{\left(5 a \right)} + \sin{\left(5 a + \frac{\pi}{2} \right)}}$$ 
  ___          
\/ 2 *cos(10*a)
---------------
     /      pi\
2*cos|5*a + --|
     \      4 /
$$\frac{\sqrt{2} \cos{\left(10 a \right)}}{2 \cos{\left(5 a + \frac{\pi}{4} \right)}}$$ 
  ___    /      pi\
\/ 2 *sec|5*a + --|
         \      4 /
-------------------
    2*sec(10*a)
$$\frac{\sqrt{2} \sec{\left(5 a + \frac{\pi}{4} \right)}}{2 \sec{\left(10 a \right)}}$$ 
    ___    /pi   5*a\
2*\/ 2 *tan|-- + ---|
           \8     2 /
---------------------
         2/pi   5*a\ 
  1 + tan |-- + ---| 
          \8     2 /
$$\frac{2 \sqrt{2} \tan{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1}$$ 
                         2                      
                 -1 + cot (5*a)                 
------------------------------------------------
                /        2/5*a\          /5*a\ \
                |-1 + cot |---|     2*cot|---| |
/       2     \ |         \ 2 /          \ 2 / |
\1 + cot (5*a)/*|-------------- - -------------|
                |       2/5*a\           2/5*a\|
                |1 + cot |---|    1 + cot |---||
                \        \ 2 /            \ 2 //
$$\frac{\cot^{2}{\left(5 a \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 a \right)} + 1\right)}$$ 
                       ___                               
                    -\/ 2 *cos(10*a)                     
---------------------------------------------------------
                      ___________    ___________         
    ___              /       ___    /       ___          
- \/ 2 *cos(5*a) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(5*a)
$$- \frac{\sqrt{2} \cos{\left(10 a \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(5 a \right)} - \sqrt{2} \cos{\left(5 a \right)}}$$ 
  ___              /      pi\
\/ 2 *cos(10*a)*sec|5*a + --|
                   \      4 /
-----------------------------
              2
$$\frac{\sqrt{2} \cos{\left(10 a \right)} \sec{\left(5 a + \frac{\pi}{4} \right)}}{2}$$ 
    ___    2/pi + 20*a\    /pi   5*a\
2*\/ 2 *sin |---------|*cot|-- + ---|
            \    8    /    \8     2 /
$$2 \sqrt{2} \sin^{2}{\left(\frac{20 a + \pi}{8} \right)} \cot{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}$$ 
2*sqrt(2)*sin((pi + 20*a)/8)^2*cot(pi/8 + 5*a/2)
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