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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
$$\frac{\cos{\left(10 a \right)}}{- a \sin{\left(5 \right)} + \cos{\left(5 a \right)}}$$ 
cos(10*a)/(cos(5*a) - sin(5)*a)
Simplificación general [src] 
    -cos(10*a)      
--------------------
-cos(5*a) + a*sin(5)
$$- \frac{\cos{\left(10 a \right)}}{a \sin{\left(5 \right)} - \cos{\left(5 a \right)}}$$ 
-cos(10*a)/(-cos(5*a) + a*sin(5))
Potencias [src] 
            -10*I*a    10*I*a          
           e          e                
           -------- + -------          
              2          2             
---------------------------------------
 -5*I*a    5*I*a       /   -5*I    5*I\
e         e        I*a*\- e     + e   /
------- + ------ + --------------------
   2        2               2
$$\frac{\frac{e^{10 i a}}{2} + \frac{e^{- 10 i a}}{2}}{\frac{i a \left(- e^{- 5 i} + e^{5 i}\right)}{2} + \frac{e^{5 i a}}{2} + \frac{e^{- 5 i a}}{2}}$$ 
(exp(-10*i*a)/2 + exp(10*i*a)/2)/(exp(-5*i*a)/2 + exp(5*i*a)/2 + i*a*(-exp(-5*i) + exp(5*i))/2)
Combinatoria [src] 
    -cos(10*a)      
--------------------
-cos(5*a) + a*sin(5)
$$- \frac{\cos{\left(10 a \right)}}{a \sin{\left(5 \right)} - \cos{\left(5 a \right)}}$$ 
-cos(10*a)/(-cos(5*a) + a*sin(5))
Denominador común [src] 
    -cos(10*a)      
--------------------
-cos(5*a) + a*sin(5)
$$- \frac{\cos{\left(10 a \right)}}{a \sin{\left(5 \right)} - \cos{\left(5 a \right)}}$$ 
-cos(10*a)/(-cos(5*a) + a*sin(5))
Parte trigonométrica [src] 
        cos(10*a)         
--------------------------
       /    pi\           
- a*cos|5 - --| + cos(5*a)
       \    2 /
$$\frac{\cos{\left(10 a \right)}}{- a \cos{\left(5 - \frac{\pi}{2} \right)} + \cos{\left(5 a \right)}}$$ 
                -1                  
------------------------------------
/     1            a     \          
|- -------- + -----------|*sec(10*a)
|  sec(5*a)      /    pi\|          
|             sec|5 - --||          
\                \    2 //
$$- \frac{1}{\left(\frac{a}{\sec{\left(5 - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
                1                 
----------------------------------
/   1            a     \          
|-------- - -----------|*sec(10*a)
|sec(5*a)      /    pi\|          
|           sec|5 - --||          
\              \    2 //
$$\frac{1}{\left(- \frac{a}{\sec{\left(5 - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
                 /       2     \                 
                -\1 - tan (5*a)/                 
-------------------------------------------------
                /         2/5*a\                \
                |  1 - tan |---|                |
/       2     \ |          \ 2 /    2*a*tan(5/2)|
\1 + tan (5*a)/*|- ------------- + -------------|
                |         2/5*a\          2     |
                |  1 + tan |---|   1 + tan (5/2)|
                \          \ 2 /                /
$$- \frac{1 - \tan^{2}{\left(5 a \right)}}{\left(\frac{2 a \tan{\left(\frac{5}{2} \right)}}{\tan^{2}{\left(\frac{5}{2} \right)} + 1} - \frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)}$$ 
                 /        2     \                 
                -\-1 + cot (5*a)/                 
--------------------------------------------------
                /          2/5*a\                \
                |  -1 + cot |---|                |
/       2     \ |           \ 2 /    2*a*cot(5/2)|
\1 + cot (5*a)/*|- -------------- + -------------|
                |         2/5*a\           2     |
                |  1 + cot |---|    1 + cot (5/2)|
                \          \ 2 /                 /
$$- \frac{\cot^{2}{\left(5 a \right)} - 1}{\left(\frac{2 a \cot{\left(\frac{5}{2} \right)}}{1 + \cot^{2}{\left(\frac{5}{2} \right)}} - \frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 a \right)} + 1\right)}$$ 
         /pi       \      
     -sin|-- + 10*a|      
         \2        /      
--------------------------
     /pi      \           
- sin|-- + 5*a| + a*sin(5)
     \2       /
$$- \frac{\sin{\left(10 a + \frac{\pi}{2} \right)}}{a \sin{\left(5 \right)} - \sin{\left(5 a + \frac{\pi}{2} \right)}}$$ 
                        2                      
                 1 - tan (5*a)                 
-----------------------------------------------
                /       2/5*a\                \
                |1 - tan |---|                |
/       2     \ |        \ 2 /    2*a*tan(5/2)|
\1 + tan (5*a)/*|------------- - -------------|
                |       2/5*a\          2     |
                |1 + tan |---|   1 + tan (5/2)|
                \        \ 2 /                /
$$\frac{1 - \tan^{2}{\left(5 a \right)}}{\left(- \frac{2 a \tan{\left(\frac{5}{2} \right)}}{\tan^{2}{\left(\frac{5}{2} \right)} + 1} + \frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(5 a \right)} + 1\right)}$$ 
         /pi       \     
      sin|-- + 10*a|     
         \2        /     
-------------------------
               /pi      \
-a*sin(5) + sin|-- + 5*a|
               \2       /
$$\frac{\sin{\left(10 a + \frac{\pi}{2} \right)}}{- a \sin{\left(5 \right)} + \sin{\left(5 a + \frac{\pi}{2} \right)}}$$ 
    -cos(10*a)      
--------------------
-cos(5*a) + a*sin(5)
$$- \frac{\cos{\left(10 a \right)}}{a \sin{\left(5 \right)} - \cos{\left(5 a \right)}}$$ 
                   1                   
---------------------------------------
/      1           a   \    /pi       \
|------------- - ------|*csc|-- - 10*a|
|   /pi      \   csc(5)|    \2        /
|csc|-- - 5*a|         |               
\   \2       /         /
$$\frac{1}{\left(- \frac{a}{\csc{\left(5 \right)}} + \frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}}\right) \csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$ 
                   -1                    
-----------------------------------------
/        1           a   \    /pi       \
|- ------------- + ------|*csc|-- - 10*a|
|     /pi      \   csc(5)|    \2        /
|  csc|-- - 5*a|         |               
\     \2       /         /
$$- \frac{1}{\left(\frac{a}{\csc{\left(5 \right)}} - \frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}}\right) \csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$ 
              1              
-----------------------------
/   1         a   \          
|-------- - ------|*sec(10*a)
\sec(5*a)   csc(5)/
$$\frac{1}{\left(- \frac{a}{\csc{\left(5 \right)}} + \frac{1}{\sec{\left(5 a \right)}}\right) \sec{\left(10 a \right)}}$$ 
       -cos(10*a)        
-------------------------
                 /    pi\
-cos(5*a) + a*cos|5 - --|
                 \    2 /
$$- \frac{\cos{\left(10 a \right)}}{a \cos{\left(5 - \frac{\pi}{2} \right)} - \cos{\left(5 a \right)}}$$ 
                         2                      
                 -1 + cot (5*a)                 
------------------------------------------------
                /        2/5*a\                \
                |-1 + cot |---|                |
/       2     \ |         \ 2 /    2*a*cot(5/2)|
\1 + cot (5*a)/*|-------------- - -------------|
                |       2/5*a\           2     |
                |1 + cot |---|    1 + cot (5/2)|
                \        \ 2 /                 /
$$\frac{\cot^{2}{\left(5 a \right)} - 1}{\left(- \frac{2 a \cot{\left(\frac{5}{2} \right)}}{1 + \cot^{2}{\left(\frac{5}{2} \right)}} + \frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(5 a \right)} + 1\right)}$$ 
(-1 + cot(5*a)^2)/((1 + cot(5*a)^2)*((-1 + cot(5*a/2)^2)/(1 + cot(5*a/2)^2) - 2*a*cot(5/2)/(1 + cot(5/2)^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) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)   - 20*cos (a) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)   - 20*cos (a) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)   - 20*cos (a) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)   - 20*cos (a) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)   - 20*cos (a) + 5*cos(a) + 16*cos (a) - 16*a*sin (1) - 5*a*sin(1) + 20*a*sin (1)
$$\frac{512 \cos^{10}{\left(a \right)}}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{1280 \cos^{8}{\left(a \right)}}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} + \frac{1120 \cos^{6}{\left(a \right)}}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{400 \cos^{4}{\left(a \right)}}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} + \frac{50 \cos^{2}{\left(a \right)}}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}} - \frac{1}{- 16 a \sin^{5}{\left(1 \right)} - 5 a \sin{\left(1 \right)} + 20 a \sin^{3}{\left(1 \right)} + 16 \cos^{5}{\left(a \right)} - 20 \cos^{3}{\left(a \right)} + 5 \cos{\left(a \right)}}$$ 
-1/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3) - 1280*cos(a)^8/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3) - 400*cos(a)^4/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3) + 50*cos(a)^2/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3) + 512*cos(a)^10/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3) + 1120*cos(a)^6/(-20*cos(a)^3 + 5*cos(a) + 16*cos(a)^5 - 16*a*sin(1)^5 - 5*a*sin(1) + 20*a*sin(1)^3)
Respuesta numérica [src] 
cos(10*a)/(0.958924274663138*a + cos(5*a))
cos(10*a)/(0.958924274663138*a + cos(5*a))
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