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Simplificación de expresiones/ Descomposición en fracciones simples/ tg4actg4a-(1-cos^9a)/(cos^3a+cos7a)

¿Cómo vas a descomponer esta tg4actg4a-(1-cos^9a)/(cos^3a+cos7a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                              9       
                       1 - cos (a)    
tan(4*a)*cot(4*a) - ------------------
                       3              
                    cos (a) + cos(7*a)
$$- \frac{1 - \cos^{9}{\left(a \right)}}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}} + \tan{\left(4 a \right)} \cot{\left(4 a \right)}$$ 
tan(4*a)*cot(4*a) - (1 - cos(a)^9)/(cos(a)^3 + cos(7*a))
Simplificación general [src] 
        3         9              
-1 + cos (a) + cos (a) + cos(7*a)
---------------------------------
           3                     
        cos (a) + cos(7*a)
$$\frac{\cos^{9}{\left(a \right)} + \cos^{3}{\left(a \right)} + \cos{\left(7 a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^3 + cos(a)^9 + cos(7*a))/(cos(a)^3 + cos(7*a))
Denominador común [src] 
           9                          
   -1 + cos (a)                       
------------------ + cot(4*a)*tan(4*a)
   3                                  
cos (a) + cos(7*a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{\cos^{9}{\left(a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^9)/(cos(a)^3 + cos(7*a)) + cot(4*a)*tan(4*a)
Potencias [src] 
                           9                                          
             / I*a    -I*a\                                           
             |e      e    |                                           
         1 - |---- + -----|              /   4*I*a    -4*I*a\         
             \ 2       2  /            I*\- e      + e      /*cot(4*a)
- ---------------------------------- + -------------------------------
                3                               -4*I*a    4*I*a       
  / I*a    -I*a\     -7*I*a    7*I*a           e       + e            
  |e      e    |    e         e                                       
  |---- + -----|  + ------- + ------                                  
  \ 2       2  /       2        2
$$- \frac{1 - \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{9}}{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{3} + \frac{e^{7 i a}}{2} + \frac{e^{- 7 i a}}{2}} + \frac{i \left(- e^{4 i a} + e^{- 4 i a}\right) \cot{\left(4 a \right)}}{e^{4 i a} + e^{- 4 i a}}$$ 
           9                          
   -1 + cos (a)                       
------------------ + cot(4*a)*tan(4*a)
   3                                  
cos (a) + cos(7*a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{\cos^{9}{\left(a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^9)/(cos(a)^3 + cos(7*a)) + cot(4*a)*tan(4*a)
Respuesta numérica [src] 
cot(4*a)*tan(4*a) - (1.0 - cos(a)^9)/(cos(a)^3 + cos(7*a))
cot(4*a)*tan(4*a) - (1.0 - cos(a)^9)/(cos(a)^3 + cos(7*a))
Unión de expresiones racionales [src] 
        9      /   3              \                  
-1 + cos (a) + \cos (a) + cos(7*a)/*cot(4*a)*tan(4*a)
-----------------------------------------------------
                     3                               
                  cos (a) + cos(7*a)
$$\frac{\left(\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} + \cos^{9}{\left(a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^9 + (cos(a)^3 + cos(7*a))*cot(4*a)*tan(4*a))/(cos(a)^3 + cos(7*a))
Denominador racional [src] 
        9      /   3              \                  
-1 + cos (a) + \cos (a) + cos(7*a)/*cot(4*a)*tan(4*a)
-----------------------------------------------------
                     3                               
                  cos (a) + cos(7*a)
$$\frac{\left(\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} + \cos^{9}{\left(a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^9 + (cos(a)^3 + cos(7*a))*cot(4*a)*tan(4*a))/(cos(a)^3 + cos(7*a))
Abrimos la expresión [src] 
                                                                               9                                                                               3                                                                                                                                                                                                            2                                                                                                      4       3                                                                                                4                                                                                                       2       3                                            
                          1                                                 cos (a)                                                                       4*tan (a)                                                                                                4*tan(a)                                                                                           24*cot (a)*tan(a)                                                                                       4*cot (a)*tan (a)                                                                                        4*cot (a)*tan(a)                                                                                       24*cot (a)*tan (a)                                         
- -------------------------------------------------- + -------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------
           5                       3            7               5                       3            7                       3            3       2           4                  3       4            2                              3            3       2           4                  3       4            2                              3            3       2           4                  3       4            2                              3            3       2           4                  3       4            2                              3            3       2           4                  3       4            2                              3            3       2           4                  3       4            2          
  - 112*cos (a) - 7*cos(a) + 57*cos (a) + 64*cos (a)   - 112*cos (a) - 7*cos(a) + 57*cos (a) + 64*cos (a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)   -4*cot(a) + 4*cot (a) - 24*cot (a)*tan (a) - 4*tan (a)*cot(a) + 4*cot (a)*tan (a) + 24*tan (a)*cot(a)
$$- \frac{4 \tan^{3}{\left(a \right)} \cot^{4}{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} + \frac{24 \tan^{3}{\left(a \right)} \cot^{2}{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} - \frac{4 \tan^{3}{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} + \frac{4 \tan{\left(a \right)} \cot^{4}{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} - \frac{24 \tan{\left(a \right)} \cot^{2}{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} + \frac{4 \tan{\left(a \right)}}{4 \tan^{4}{\left(a \right)} \cot^{3}{\left(a \right)} - 4 \tan^{4}{\left(a \right)} \cot{\left(a \right)} - 24 \tan^{2}{\left(a \right)} \cot^{3}{\left(a \right)} + 24 \tan^{2}{\left(a \right)} \cot{\left(a \right)} + 4 \cot^{3}{\left(a \right)} - 4 \cot{\left(a \right)}} + \frac{\cos^{9}{\left(a \right)}}{64 \cos^{7}{\left(a \right)} - 112 \cos^{5}{\left(a \right)} + 57 \cos^{3}{\left(a \right)} - 7 \cos{\left(a \right)}} - \frac{1}{64 \cos^{7}{\left(a \right)} - 112 \cos^{5}{\left(a \right)} + 57 \cos^{3}{\left(a \right)} - 7 \cos{\left(a \right)}}$$ 
-1/(-112*cos(a)^5 - 7*cos(a) + 57*cos(a)^3 + 64*cos(a)^7) + cos(a)^9/(-112*cos(a)^5 - 7*cos(a) + 57*cos(a)^3 + 64*cos(a)^7) - 4*tan(a)^3/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a)) + 4*tan(a)/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a)) - 24*cot(a)^2*tan(a)/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a)) - 4*cot(a)^4*tan(a)^3/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a)) + 4*cot(a)^4*tan(a)/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a)) + 24*cot(a)^2*tan(a)^3/(-4*cot(a) + 4*cot(a)^3 - 24*cot(a)^3*tan(a)^2 - 4*tan(a)^4*cot(a) + 4*cot(a)^3*tan(a)^4 + 24*tan(a)^2*cot(a))
Combinatoria [src] 
        9         3                                                  
-1 + cos (a) + cos (a)*cot(4*a)*tan(4*a) + cos(7*a)*cot(4*a)*tan(4*a)
---------------------------------------------------------------------
                             3                                       
                          cos (a) + cos(7*a)
$$\frac{\cos^{9}{\left(a \right)} + \cos^{3}{\left(a \right)} \tan{\left(4 a \right)} \cot{\left(4 a \right)} + \cos{\left(7 a \right)} \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{\cos^{3}{\left(a \right)} + \cos{\left(7 a \right)}}$$ 
(-1 + cos(a)^9 + cos(a)^3*cot(4*a)*tan(4*a) + cos(7*a)*cot(4*a)*tan(4*a))/(cos(a)^3 + cos(7*a))
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