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Simplificación de expresiones/ Descomposición en fracciones simples/ tg4actg4a-(1-(cos^2)9a)/((sin^2)9a-1)

¿Cómo vas a descomponer esta tg4actg4a-(1-(cos^2)9a)/((sin^2)9a-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                           2       
                    1 - cos (a)*9*a
tan(4*a)*cot(4*a) - ---------------
                       2           
                    sin (a)*9*a - 1
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} - \frac{- a 9 \cos^{2}{\left(a \right)} + 1}{a 9 \sin^{2}{\left(a \right)} - 1}$$ 
tan(4*a)*cot(4*a) - (1 - cos(a)^2*9*a)/((sin(a)^2*9)*a - 1)
Simplificación general [src] 
    -2 + 9*a    
----------------
            2   
-1 + 9*a*sin (a)
$$\frac{9 a - 2}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
(-2 + 9*a)/(-1 + 9*a*sin(a)^2)
Respuesta numérica [src] 
cot(4*a)*tan(4*a) - (1.0 - 9.0*a*cos(a)^2)/(-1.0 + 9.0*a*sin(a)^2)
cot(4*a)*tan(4*a) - (1.0 - 9.0*a*cos(a)^2)/(-1.0 + 9.0*a*sin(a)^2)
Denominador racional [src] 
            2      /            2   \                  
-1 + 9*a*cos (a) + \-1 + 9*a*sin (a)/*cot(4*a)*tan(4*a)
-------------------------------------------------------
                                2                      
                    -1 + 9*a*sin (a)
$$\frac{9 a \cos^{2}{\left(a \right)} + \left(9 a \sin^{2}{\left(a \right)} - 1\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
(-1 + 9*a*cos(a)^2 + (-1 + 9*a*sin(a)^2)*cot(4*a)*tan(4*a))/(-1 + 9*a*sin(a)^2)
Unión de expresiones racionales [src] 
            2      /            2   \                  
-1 + 9*a*cos (a) + \-1 + 9*a*sin (a)/*cot(4*a)*tan(4*a)
-------------------------------------------------------
                                2                      
                    -1 + 9*a*sin (a)
$$\frac{9 a \cos^{2}{\left(a \right)} + \left(9 a \sin^{2}{\left(a \right)} - 1\right) \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
(-1 + 9*a*cos(a)^2 + (-1 + 9*a*sin(a)^2)*cot(4*a)*tan(4*a))/(-1 + 9*a*sin(a)^2)
Potencias [src] 
            2                       
-1 + 9*a*cos (a)                    
---------------- + cot(4*a)*tan(4*a)
            2                       
-1 + 9*a*sin (a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{9 a \cos^{2}{\left(a \right)} - 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
                         2                                    
           / I*a    -I*a\                                     
           |e      e    |                                     
   1 - 9*a*|---- + -----|        /   4*I*a    -4*I*a\         
           \ 2       2  /      I*\- e      + e      /*cot(4*a)
- -------------------------- + -------------------------------
                           2            -4*I*a    4*I*a       
           /   -I*a    I*a\            e       + e            
       9*a*\- e     + e   /                                   
  -1 - ---------------------                                  
                 4
$$- \frac{- 9 a \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} + 1}{- \frac{9 a \left(e^{i a} - e^{- i a}\right)^{2}}{4} - 1} + \frac{i \left(- e^{4 i a} + e^{- 4 i a}\right) \cot{\left(4 a \right)}}{e^{4 i a} + e^{- 4 i a}}$$ 
                               2    
                    1 - 9*a*cos (a) 
cot(4*a)*tan(4*a) - ----------------
                                2   
                    -1 + 9*a*sin (a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} - \frac{- 9 a \cos^{2}{\left(a \right)} + 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
cot(4*a)*tan(4*a) - (1 - 9*a*cos(a)^2)/(-1 + 9*a*sin(a)^2)
Combinatoria [src] 
                                2             2                     
-1 - cot(4*a)*tan(4*a) + 9*a*cos (a) + 9*a*sin (a)*cot(4*a)*tan(4*a)
--------------------------------------------------------------------
                                      2                             
                          -1 + 9*a*sin (a)
$$\frac{9 a \sin^{2}{\left(a \right)} \tan{\left(4 a \right)} \cot{\left(4 a \right)} + 9 a \cos^{2}{\left(a \right)} - \tan{\left(4 a \right)} \cot{\left(4 a \right)} - 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
(-1 - cot(4*a)*tan(4*a) + 9*a*cos(a)^2 + 9*a*sin(a)^2*cot(4*a)*tan(4*a))/(-1 + 9*a*sin(a)^2)
Compilar la expresión [src] 
                               2    
                    1 - 9*a*cos (a) 
cot(4*a)*tan(4*a) - ----------------
                                2   
                    -1 + 9*a*sin (a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} - \frac{- 9 a \cos^{2}{\left(a \right)} + 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
cot(4*a)*tan(4*a) - (1 - 9*a*cos(a)^2)/(-1 + 9*a*sin(a)^2)
Abrimos la expresión [src] 
                                                                       3                                                                                                                                                                                                            2                                                                                                      4       3                                                                                                4                                                                2                                                        2       3                                            
         1                                                        4*tan (a)                                                                                                4*tan(a)                                                                                           24*cot (a)*tan(a)                                                                                       4*cot (a)*tan (a)                                                                                        4*cot (a)*tan(a)                                               9*a*cos (a)                                               24*cot (a)*tan (a)                                         
- --------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- + --------------- + -----------------------------------------------------------------------------------------------------
     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                2                               3            3       2           4                  3       4            2          
  sin (a)*9*a - 1   -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)   sin (a)*9*a - 1   -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{9 a \cos^{2}{\left(a \right)}}{a 9 \sin^{2}{\left(a \right)} - 1} - \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{1}{a 9 \sin^{2}{\left(a \right)} - 1}$$ 
-1/((sin(a)^2*9)*a - 1) - 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)) + 9*a*cos(a)^2/((sin(a)^2*9)*a - 1) + 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))
Denominador común [src] 
            2                       
-1 + 9*a*cos (a)                    
---------------- + cot(4*a)*tan(4*a)
            2                       
-1 + 9*a*sin (a)
$$\tan{\left(4 a \right)} \cot{\left(4 a \right)} + \frac{9 a \cos^{2}{\left(a \right)} - 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
(-1 + 9*a*cos(a)^2)/(-1 + 9*a*sin(a)^2) + cot(4*a)*tan(4*a)
Parte trigonométrica [src] 
          9*a   
    1 - ------- 
           2    
        sec (a) 
1 - ------------
           9*a  
    -1 + -------
            2   
         csc (a)
$$1 - \frac{- \frac{9 a}{\sec^{2}{\left(a \right)}} + 1}{\frac{9 a}{\csc^{2}{\left(a \right)}} - 1}$$ 
      -2 + 9*a     
-------------------
              2/a\ 
      36*a*tan |-| 
               \2/ 
-1 + --------------
                  2
     /       2/a\\ 
     |1 + tan |-|| 
     \        \2//
$$\frac{9 a - 2}{\frac{36 a \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - 1}$$ 
                         2
            /       2/a\\ 
        9*a*|1 - tan |-|| 
            \        \2// 
    1 - ------------------
                       2  
          /       2/a\\   
          |1 + tan |-||   
          \        \2//   
1 - ----------------------
                   2/a\   
           36*a*tan |-|   
                    \2/   
     -1 + --------------  
                       2  
          /       2/a\\   
          |1 + tan |-||   
          \        \2//
$$- \frac{- \frac{9 a \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1}{\frac{36 a \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - 1} + 1$$ 
    -2 + 9*a    
----------------
            2   
-1 + 9*a*sin (a)
$$\frac{9 a - 2}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
       -2 + 9*a      
---------------------
            2/    pi\
-1 + 9*a*cos |a - --|
             \    2 /
$$\frac{9 a - 2}{9 a \cos^{2}{\left(a - \frac{\pi}{2} \right)} - 1}$$ 
      -2 + 9*a     
-------------------
              2/a\ 
      36*a*cot |-| 
               \2/ 
-1 + --------------
                  2
     /       2/a\\ 
     |1 + cot |-|| 
     \        \2//
$$\frac{9 a - 2}{\frac{36 a \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - 1}$$ 
               2/    pi\
    1 - 9*a*sin |a + --|
                \    2 /
1 - --------------------
                  2     
      -1 + 9*a*sin (a)
$$1 - \frac{- 9 a \sin^{2}{\left(a + \frac{\pi}{2} \right)} + 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
             9*a     
       1 - -------   
              2      
           sec (a)   
1 - -----------------
             9*a     
    -1 + ------------
            2/    pi\
         sec |a - --|
             \    2 /
$$- \frac{- \frac{9 a}{\sec^{2}{\left(a \right)}} + 1}{\frac{9 a}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - 1} + 1$$ 
  -2 + 9*a  
------------
       9*a  
-1 + -------
        2   
     csc (a)
$$\frac{9 a - 2}{\frac{9 a}{\csc^{2}{\left(a \right)}} - 1}$$ 
               2    
    1 - 9*a*cos (a) 
1 - ----------------
                2   
    -1 + 9*a*sin (a)
$$1 - \frac{- 9 a \cos^{2}{\left(a \right)} + 1}{9 a \sin^{2}{\left(a \right)} - 1}$$ 
            9*a     
    1 - ------------
           2/pi    \
        csc |-- - a|
            \2     /
1 - ----------------
             9*a    
      -1 + -------  
              2     
           csc (a)
$$1 - \frac{- \frac{9 a}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1}{\frac{9 a}{\csc^{2}{\left(a \right)}} - 1}$$ 
     -2 + 9*a    
-----------------
         9*a     
-1 + ------------
        2/    pi\
     sec |a - --|
         \    2 /
$$\frac{9 a - 2}{\frac{9 a}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}$$ 
                          2
            /        2/a\\ 
        9*a*|-1 + cot |-|| 
            \         \2// 
    1 - -------------------
                        2  
           /       2/a\\   
           |1 + cot |-||   
           \        \2//   
1 - -----------------------
                    2/a\   
            36*a*cot |-|   
                     \2/   
      -1 + --------------  
                        2  
           /       2/a\\   
           |1 + cot |-||   
           \        \2//
$$- \frac{- \frac{9 a \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1}{\frac{36 a \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - 1} + 1$$ 
                  2      
       1 - 9*a*cos (a)   
1 - ---------------------
                2/    pi\
    -1 + 9*a*cos |a - --|
                 \    2 /
$$- \frac{- 9 a \cos^{2}{\left(a \right)} + 1}{9 a \cos^{2}{\left(a - \frac{\pi}{2} \right)} - 1} + 1$$ 
1 - (1 - 9*a*cos(a)^2)/(-1 + 9*a*cos(a - pi/2)^2)
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