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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
     cos(10*a)     
-------------------
sin(5*a) - cos(5*a)
$$\frac{\cos{\left(10 a \right)}}{\sin{\left(5 a \right)} - \cos{\left(5 a \right)}}$$ 
cos(10*a)/(sin(5*a) - cos(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)
Combinatoria [src] 
    -cos(10*a)      
--------------------
-sin(5*a) + cos(5*a)
$$- \frac{\cos{\left(10 a \right)}}{- \sin{\left(5 a \right)} + \cos{\left(5 a \right)}}$$ 
-cos(10*a)/(-sin(5*a) + cos(5*a))
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*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (a)   - 20*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (a)   - 20*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (a)   - 20*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (a)   - 20*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (a)   - 20*sin (a) - 16*cos (a) - 5*cos(a) + 5*sin(a) + 16*sin (a) + 20*cos (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*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3) - 1280*cos(a)^8/(-20*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3) - 400*cos(a)^4/(-20*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3) + 50*cos(a)^2/(-20*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3) + 512*cos(a)^10/(-20*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3) + 1120*cos(a)^6/(-20*sin(a)^3 - 16*cos(a)^5 - 5*cos(a) + 5*sin(a) + 16*sin(a)^5 + 20*cos(a)^3)
Respuesta numérica [src] 
cos(10*a)/(-cos(5*a) + sin(5*a))
cos(10*a)/(-cos(5*a) + sin(5*a))
Parte trigonométrica [src] 
                    1                    
-----------------------------------------
/   1             1      \    /pi       \
|-------- - -------------|*csc|-- - 10*a|
|csc(5*a)      /pi      \|    \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                       
                  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 //
$$\frac{1 - \tan^{2}{\left(5 a \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)}$$ 
     ___    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                       
                  -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 //
$$\frac{\cot^{2}{\left(5 a \right)} - 1}{\left(- \frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 a \right)} + 1\right)}$$ 
   ___    /      pi\
-\/ 2 *cos|5*a - --|
          \      4 /
$$- \sqrt{2} \cos{\left(5 a - \frac{\pi}{4} \right)}$$ 
   ___    /      pi\ 
-\/ 2 *sec|5*a + --| 
          \      4 / 
---------------------
     2*sec(10*a)
$$- \frac{\sqrt{2} \sec{\left(5 a + \frac{\pi}{4} \right)}}{2 \sec{\left(10 a \right)}}$$ 
   ___    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 //
$$- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(5 a \right)}\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(5 a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}\right)}$$ 
   ___              /       pi\ 
-\/ 2 *cos(10*a)*csc|-5*a + --| 
                    \       4 / 
--------------------------------
               2
$$- \frac{\sqrt{2} \cos{\left(10 a \right)} \csc{\left(- 5 a + \frac{\pi}{4} \right)}}{2}$$ 
     ___    /pi   5*a\
-2*\/ 2 *cot|-- + ---|
            \8     2 /
----------------------
         2/pi   5*a\  
  1 + cot |-- + ---|  
          \8     2 /
$$- \frac{2 \sqrt{2} \cot{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1}$$ 
   ___ /       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 *cos(10*a)                     
---------------------------------------------------------
                      ___________    ___________         
    ___              /       ___    /       ___          
- \/ 2 *cos(5*a) + \/  2 + \/ 2  *\/  2 - \/ 2  *sin(5*a)
$$\frac{\sqrt{2} \cos{\left(10 a \right)}}{\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(5 a \right)} - \sqrt{2} \cos{\left(5 a \right)}}$$ 
   ___           
-\/ 2 *cos(10*a) 
-----------------
      /      pi\ 
 2*cos|5*a + --| 
      \      4 /
$$- \frac{\sqrt{2} \cos{\left(10 a \right)}}{2 \cos{\left(5 a + \frac{\pi}{4} \right)}}$$ 
      ___    
   -\/ 2     
-------------
   /      pi\
csc|5*a + --|
   \      4 /
$$- \frac{\sqrt{2}}{\csc{\left(5 a + \frac{\pi}{4} \right)}}$$ 
                 1                  
------------------------------------
/      1            1    \          
|------------- - --------|*sec(10*a)
|   /      pi\   sec(5*a)|          
|sec|5*a - --|           |          
\   \      2 /           /
$$\frac{1}{\left(\frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
   ___    /       pi\ 
-\/ 2 *csc|-5*a + --| 
          \       4 / 
----------------------
        /pi       \   
   2*csc|-- - 10*a|   
        \2        /
$$- \frac{\sqrt{2} \csc{\left(- 5 a + \frac{\pi}{4} \right)}}{2 \csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$ 
               1               
-------------------------------
/   1          1    \          
|-------- - --------|*sec(10*a)
\csc(5*a)   sec(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/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 //
$$- \frac{\sqrt{2} \left(\cot^{2}{\left(5 a \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(\cot^{2}{\left(5 a \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} - 1\right)}$$ 
     ___    /pi   5*a\
-2*\/ 2 *tan|-- + ---|
            \8     2 /
----------------------
         2/pi   5*a\  
  1 + tan |-- + ---|  
          \8     2 /
$$- \frac{2 \sqrt{2} \tan{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{8} \right)} + 1}$$ 
         /pi       \      
      sin|-- + 10*a|      
         \2        /      
--------------------------
     /pi      \           
- sin|-- + 5*a| + sin(5*a)
     \2       /
$$\frac{\sin{\left(10 a + \frac{\pi}{2} \right)}}{\sin{\left(5 a \right)} - \sin{\left(5 a + \frac{\pi}{2} \right)}}$$ 
        cos(10*a)        
-------------------------
               /      pi\
-cos(5*a) + cos|5*a - --|
               \      2 /
$$\frac{\cos{\left(10 a \right)}}{- \cos{\left(5 a \right)} + \cos{\left(5 a - \frac{\pi}{2} \right)}}$$ 
   ___              /      pi\ 
-\/ 2 *cos(10*a)*sec|5*a + --| 
                    \      4 / 
-------------------------------
               2
$$- \frac{\sqrt{2} \cos{\left(10 a \right)} \sec{\left(5 a + \frac{\pi}{4} \right)}}{2}$$ 
   ___    /      pi\
-\/ 2 *sin|5*a + --|
          \      4 /
$$- \sqrt{2} \sin{\left(5 a + \frac{\pi}{4} \right)}$$ 
   ___    /pi       \ 
-\/ 2 *sin|-- + 10*a| 
          \2        / 
----------------------
       /      3*pi\   
  2*sin|5*a + ----|   
       \       4  /
$$- \frac{\sqrt{2} \sin{\left(10 a + \frac{\pi}{2} \right)}}{2 \sin{\left(5 a + \frac{3 \pi}{4} \right)}}$$ 
      ___    
   -\/ 2     
-------------
   /      pi\
sec|5*a - --|
   \      4 /
$$- \frac{\sqrt{2}}{\sec{\left(5 a - \frac{\pi}{4} \right)}}$$ 
-sqrt(2)/sec(5*a - pi/4)
Denominador común [src] 
    -cos(10*a)      
--------------------
-sin(5*a) + cos(5*a)
$$- \frac{\cos{\left(10 a \right)}}{- \sin{\left(5 a \right)} + \cos{\left(5 a \right)}}$$ 
-cos(10*a)/(-sin(5*a) + cos(5*a))
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