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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(6*a)/sin(2*a)+sin(6*a)/((cos(2)*a))

¿Cómo vas a descomponer esta cos(6*a)/sin(2*a)+sin(6*a)/((cos(2)*a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(6*a)   sin(6*a)
-------- + --------
sin(2*a)   cos(2)*a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)}}{a \cos{\left(2 \right)}}$$ 
cos(6*a)/sin(2*a) + sin(6*a)/((cos(2)*a))
Respuesta numérica [src] 
cos(6*a)/sin(2*a) - 2.40299796172238*sin(6*a)/a
cos(6*a)/sin(2*a) - 2.40299796172238*sin(6*a)/a
Combinatoria [src] 
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
          a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$ 
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Denominador racional [src] 
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
          a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$ 
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Unión de expresiones racionales [src] 
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
          a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$ 
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Denominador común [src] 
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
          a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$ 
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Potencias [src] 
    / -6*I*a    6*I*a\                         
    |e         e     |                         
2*I*|------- + ------|     /   -6*I*a    6*I*a\
    \   2        2   /   I*\- e       + e     /
---------------------- - ----------------------
     -2*I*a    2*I*a           / -2*I    2*I\  
  - e       + e                |e       e   |  
                           2*a*|----- + ----|  
                               \  2      2  /
$$\frac{2 i \left(\frac{e^{6 i a}}{2} + \frac{e^{- 6 i a}}{2}\right)}{e^{2 i a} - e^{- 2 i a}} - \frac{i \left(e^{6 i a} - e^{- 6 i a}\right)}{2 a \left(\frac{e^{- 2 i}}{2} + \frac{e^{2 i}}{2}\right)}$$ 
2*i*(exp(-6*i*a)/2 + exp(6*i*a)/2)/(-exp(-2*i*a) + exp(2*i*a)) - i*(-exp(-6*i*a) + exp(6*i*a))/(2*a*(exp(-2*i)/2 + exp(2*i)/2))
Abrimos la expresión [src] 
        3                       5                              3                                      5          
  24*cos (a)   9*cos(a)   16*cos (a)          1          32*sin (a)*cos(a)   6*cos(a)*sin(a)    32*sin (a)*cos(a)
- ---------- + -------- + ---------- - --------------- - ----------------- + ---------------- + -----------------
    sin(a)      sin(a)      sin(a)     2*cos(a)*sin(a)                2                  2                   2   
                                                          -a + 2*a*cos (1)   -a + 2*a*cos (1)    -a + 2*a*cos (1)
$$\frac{16 \cos^{5}{\left(a \right)}}{\sin{\left(a \right)}} - \frac{24 \cos^{3}{\left(a \right)}}{\sin{\left(a \right)}} + \frac{9 \cos{\left(a \right)}}{\sin{\left(a \right)}} - \frac{1}{2 \sin{\left(a \right)} \cos{\left(a \right)}} + \frac{32 \sin^{5}{\left(a \right)} \cos{\left(a \right)}}{- a + 2 a \cos^{2}{\left(1 \right)}} - \frac{32 \sin^{3}{\left(a \right)} \cos{\left(a \right)}}{- a + 2 a \cos^{2}{\left(1 \right)}} + \frac{6 \sin{\left(a \right)} \cos{\left(a \right)}}{- a + 2 a \cos^{2}{\left(1 \right)}}$$ 
-24*cos(a)^3/sin(a) + 9*cos(a)/sin(a) + 16*cos(a)^5/sin(a) - 1/(2*cos(a)*sin(a)) - 32*sin(a)^3*cos(a)/(-a + 2*a*cos(1)^2) + 6*cos(a)*sin(a)/(-a + 2*a*cos(1)^2) + 32*sin(a)^5*cos(a)/(-a + 2*a*cos(1)^2)
Parte trigonométrica [src] 
                    sec(2)*sin(6*a)
cos(6*a)*csc(2*a) + ---------------
                           a
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} + \frac{\sin{\left(6 a \right)} \sec{\left(2 \right)}}{a}$$ 
                        1                   1         
                ----------------- + ------------------
                          2                 2/     pi\
                       csc (1)           csc |-1 + --|
                1 - -------------            \     2 /
                       2/     pi\   -1 + -------------
                    csc |-1 + --|              2      
   csc(2*a)             \     2 /           csc (1)   
------------- - --------------------------------------
   /pi      \                 a*csc(6*a)              
csc|-- - 6*a|                                         
   \2       /
$$\frac{\csc{\left(2 a \right)}}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} - \frac{\frac{1}{-1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}} + \frac{1}{- \frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1}}{a \csc{\left(6 a \right)}}$$ 
                                   /     1             1      \         
                                 2*|----------- + ------------|*cot(3*a)
                                   |       2              1   |         
                                   |1 - cot (1)   -1 + -------|         
/       2   \ /        2     \     |                      2   |         
\1 + cot (a)/*\-1 + cot (3*a)/     \                   cot (1)/         
------------------------------ - ---------------------------------------
     /       2     \                          /       2     \           
   2*\1 + cot (3*a)/*cot(a)                 a*\1 + cot (3*a)/
$$\frac{\left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} - 1\right)}{2 \left(\cot^{2}{\left(3 a \right)} + 1\right) \cot{\left(a \right)}} - \frac{2 \left(\frac{1}{-1 + \frac{1}{\cot^{2}{\left(1 \right)}}} + \frac{1}{1 - \cot^{2}{\left(1 \right)}}\right) \cot{\left(3 a \right)}}{a \left(\cot^{2}{\left(3 a \right)} + 1\right)}$$ 
csc(2*a)     sec(2)  
-------- + ----------
sec(6*a)   a*csc(6*a)
$$\frac{\csc{\left(2 a \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \csc{\left(6 a \right)}}$$ 
                                  /     1             1      \         
                                2*|----------- + ------------|*tan(3*a)
                                  |       1              2   |         
                                  |1 - -------   -1 + tan (1)|         
/       2   \ /       2     \     |       2                  |         
\1 + tan (a)/*\1 - tan (3*a)/     \    tan (1)               /         
----------------------------- - ---------------------------------------
     /       2     \                         /       2     \           
   2*\1 + tan (3*a)/*tan(a)                a*\1 + tan (3*a)/
$$\frac{\left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}{2 \left(\tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(a \right)}} - \frac{2 \left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1}{1 - \frac{1}{\tan^{2}{\left(1 \right)}}}\right) \tan{\left(3 a \right)}}{a \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$ 
                   /     pi\
                csc|-2 + --|
   csc(2*a)        \     2 /
------------- + ------------
   /pi      \    a*csc(6*a) 
csc|-- - 6*a|               
   \2       /
$$\frac{\csc{\left(2 a \right)}}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} + \frac{\csc{\left(-2 + \frac{\pi}{2} \right)}}{a \csc{\left(6 a \right)}}$$ 
cos(6*a)   sec(2)*sin(6*a)
-------- + ---------------
sin(2*a)          a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)} \sec{\left(2 \right)}}{a}$$ 
                    /                     1   \         
                    |               1 - ------|         
                    |     1             cos(2)|         
                    |------------ + ----------|*sin(6*a)
                    |        2          2     |         
                    \-1 + tan (1)             /         
cos(6*a)*csc(2*a) - ------------------------------------
                                     a
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \sin{\left(6 a \right)}}{a}$$ 
                /      1               1       \         
                |------------- + --------------|*sin(6*a)
                |        2                 4   |         
                |     sin (2)         4*sin (1)|         
   /pi      \   |1 - ---------   -1 + ---------|         
sin|-- + 6*a|   |         4               2    |         
   \2       /   \    4*sin (1)         sin (2) /         
------------- - -----------------------------------------
   sin(2*a)                         a
$$\frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}} + \frac{1}{- \frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1}\right) \sin{\left(6 a \right)}}{a}$$ 
           /1               2   cos(4)\         
           |- + (1 - cos(2))  - ------|*sin(6*a)
cos(6*a)   \2                     2   /         
-------- - -------------------------------------
sin(2*a)         a*(1 - 2*cos(2) + cos(4))
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(- \frac{\cos{\left(4 \right)}}{2} + \frac{1}{2} + \left(1 - \cos{\left(2 \right)}\right)^{2}\right) \sin{\left(6 a \right)}}{a \left(\cos{\left(4 \right)} - 2 \cos{\left(2 \right)} + 1\right)}$$ 
                       /                     1   \         
                       |               1 - ------|         
                2      |     1             cos(2)|         
           2*cos (3*a)*|------------ + ----------|*tan(3*a)
                       |        2          2     |         
cos(6*a)               \-1 + tan (1)             /         
-------- - ------------------------------------------------
sin(2*a)                          a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{2 \left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \cos^{2}{\left(3 a \right)} \tan{\left(3 a \right)}}{a}$$ 
   /pi      \                  
sec|-- - 2*a|                  
   \2       /        sec(2)    
------------- + ---------------
   sec(6*a)          /      pi\
                a*sec|6*a - --|
                     \      2 /
$$\frac{\sec{\left(- 2 a + \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$ 
   /pi      \                
sin|-- + 6*a|                
   \2       /      sin(6*a)  
------------- + -------------
   sin(2*a)          /    pi\
                a*sin|2 + --|
                     \    2 /
$$\frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)}}{a \sin{\left(\frac{\pi}{2} + 2 \right)}}$$ 
                   /      pi\
                cos|6*a - --|
   cos(6*a)        \      2 /
------------- + -------------
   /      pi\      a*cos(2)  
cos|2*a - --|                
   \      2 /
$$\frac{\cos{\left(6 a \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(6 a - \frac{\pi}{2} \right)}}{a \cos{\left(2 \right)}}$$ 
/       2   \ /        2     \         /       2   \             
\1 + cot (a)/*\-1 + cot (3*a)/       2*\1 + cot (1)/*cot(3*a)    
------------------------------ + --------------------------------
     /       2     \               /       2     \ /        2   \
   2*\1 + cot (3*a)/*cot(a)      a*\1 + cot (3*a)/*\-1 + cot (1)/
$$\frac{\left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} - 1\right)}{2 \left(\cot^{2}{\left(3 a \right)} + 1\right) \cot{\left(a \right)}} + \frac{2 \left(\cot^{2}{\left(1 \right)} + 1\right) \cot{\left(3 a \right)}}{a \left(-1 + \cot^{2}{\left(1 \right)}\right) \left(\cot^{2}{\left(3 a \right)} + 1\right)}$$ 
                /       1                   1        \    /      pi\
                |---------------- + -----------------|*cos|6*a - --|
                |         2                 2/    pi\|    \      2 /
                |      cos (1)           cos |1 - --||              
                |1 - ------------            \    2 /|              
                |       2/    pi\   -1 + ------------|              
                |    cos |1 - --|             2      |              
   cos(6*a)     \        \    2 /          cos (1)   /              
------------- - ----------------------------------------------------
   /      pi\                            a                          
cos|2*a - --|                                                       
   \      2 /
$$\frac{\cos{\left(6 a \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)}} - \frac{\left(\frac{1}{-1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}} + \frac{1}{- \frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}} + 1}\right) \cos{\left(6 a - \frac{\pi}{2} \right)}}{a}$$ 
           /     1             1      \         
           |----------- + ------------|*sin(6*a)
           |       1              2   |         
           |1 - -------   -1 + tan (1)|         
           |       2                  |         
cos(6*a)   \    tan (1)               /         
-------- - -------------------------------------
sin(2*a)                     a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1}{1 - \frac{1}{\tan^{2}{\left(1 \right)}}}\right) \sin{\left(6 a \right)}}{a}$$ 
                       1                   1        
                ---------------- + -----------------
                       2/    pi\             2      
                    sec |1 - --|          sec (1)   
                        \    2 /   -1 + ------------
   /      pi\   1 - ------------           2/    pi\
sec|2*a - --|            2              sec |1 - --|
   \      2 /         sec (1)               \    2 /
------------- - ------------------------------------
   sec(6*a)                    /      pi\           
                          a*sec|6*a - --|           
                               \      2 /
$$\frac{\sec{\left(2 a - \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} - \frac{\frac{1}{-1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}} + \frac{1}{- \frac{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}{\sec^{2}{\left(1 \right)}} + 1}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$ 
           /                     1   \         
           |               1 - ------|         
           |     1             cos(2)|         
           |------------ + ----------|*sin(6*a)
           |        2          2     |         
cos(6*a)   \-1 + tan (1)             /         
-------- - ------------------------------------
sin(2*a)                    a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \sin{\left(6 a \right)}}{a}$$ 
   /      pi\                  
sec|2*a - --|                  
   \      2 /        sec(2)    
------------- + ---------------
   sec(6*a)          /      pi\
                a*sec|6*a - --|
                     \      2 /
$$\frac{\sec{\left(2 a - \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$ 
                    sin(6*a)
cos(6*a)*csc(2*a) + --------
                    a*cos(2)
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} + \frac{\sin{\left(6 a \right)}}{a \cos{\left(2 \right)}}$$ 
/       2   \ /       2     \         /       2   \            
\1 + tan (a)/*\1 - tan (3*a)/       2*\1 + tan (1)/*tan(3*a)   
----------------------------- + -------------------------------
     /       2     \              /       2     \ /       2   \
   2*\1 + tan (3*a)/*tan(a)     a*\1 + tan (3*a)/*\1 - tan (1)/
$$\frac{\left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}{2 \left(\tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(a \right)}} + \frac{2 \left(1 + \tan^{2}{\left(1 \right)}\right) \tan{\left(3 a \right)}}{a \left(1 - \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$ 
(1 + tan(a)^2)*(1 - tan(3*a)^2)/(2*(1 + tan(3*a)^2)*tan(a)) + 2*(1 + tan(1)^2)*tan(3*a)/(a*(1 + tan(3*a)^2)*(1 - tan(1)^2))
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