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Simplificación de expresiones/ Descomposición en fracciones simples/ tg2α*ctg2α-(1-sin^2*5α)/(cos^2*5α-1)

¿Cómo vas a descomponer esta tg2α*ctg2α-(1-sin^2*5α)/(cos^2*5α-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                           2     
                    1 - sin (5)*a
tan(2*a)*cot(2*a) - -------------
                       2         
                    cos (5)*a - 1
$$- \frac{- a \sin^{2}{\left(5 \right)} + 1}{a \cos^{2}{\left(5 \right)} - 1} + \tan{\left(2 a \right)} \cot{\left(2 a \right)}$$ 
tan(2*a)*cot(2*a) - (1 - sin(5)^2*a)/(cos(5)^2*a - 1)
Simplificación general [src] 
    -2 + a    
--------------
          2   
-1 + a*cos (5)
$$\frac{a - 2}{a \cos^{2}{\left(5 \right)} - 1}$$ 
(-2 + a)/(-1 + a*cos(5)^2)
Denominador racional [src] 
          2      /          2   \                  
-1 + a*sin (5) + \-1 + a*cos (5)/*cot(2*a)*tan(2*a)
---------------------------------------------------
                             2                     
                   -1 + a*cos (5)
$$\frac{a \sin^{2}{\left(5 \right)} + \left(a \cos^{2}{\left(5 \right)} - 1\right) \tan{\left(2 a \right)} \cot{\left(2 a \right)} - 1}{a \cos^{2}{\left(5 \right)} - 1}$$ 
(-1 + a*sin(5)^2 + (-1 + a*cos(5)^2)*cot(2*a)*tan(2*a))/(-1 + a*cos(5)^2)
Unión de expresiones racionales [src] 
          2      /          2   \                  
-1 + a*sin (5) + \-1 + a*cos (5)/*cot(2*a)*tan(2*a)
---------------------------------------------------
                             2                     
                   -1 + a*cos (5)
$$\frac{a \sin^{2}{\left(5 \right)} + \left(a \cos^{2}{\left(5 \right)} - 1\right) \tan{\left(2 a \right)} \cot{\left(2 a \right)} - 1}{a \cos^{2}{\left(5 \right)} - 1}$$ 
(-1 + a*sin(5)^2 + (-1 + a*cos(5)^2)*cot(2*a)*tan(2*a))/(-1 + a*cos(5)^2)
Potencias [src] 
          2                       
-1 + a*sin (5)                    
-------------- + cot(2*a)*tan(2*a)
          2                       
-1 + a*cos (5)
$$\frac{a \sin^{2}{\left(5 \right)} - 1}{a \cos^{2}{\left(5 \right)} - 1} + \tan{\left(2 a \right)} \cot{\left(2 a \right)}$$ 
                        2                                  
        /   -5*I    5*I\                                   
      a*\- e     + e   /                                   
  1 + -------------------     /   2*I*a    -2*I*a\         
               4            I*\- e      + e      /*cot(2*a)
- ----------------------- + -------------------------------
                        2            -2*I*a    2*I*a       
          / -5*I    5*I\            e       + e            
          |e       e   |                                   
   -1 + a*|----- + ----|                                   
          \  2      2  /
$$- \frac{\frac{a \left(- e^{- 5 i} + e^{5 i}\right)^{2}}{4} + 1}{a \left(\frac{e^{5 i}}{2} + \frac{e^{- 5 i}}{2}\right)^{2} - 1} + \frac{i \left(- e^{2 i a} + e^{- 2 i a}\right) \cot{\left(2 a \right)}}{e^{2 i a} + e^{- 2 i a}}$$ 
-(1 + a*(-exp(-5*i) + exp(5*i))^2/4)/(-1 + a*(exp(-5*i)/2 + exp(5*i)/2)^2) + i*(-exp(2*i*a) + exp(-2*i*a))*cot(2*a)/(exp(-2*i*a) + exp(2*i*a))
Combinatoria [src] 
          2                               2                     
-1 + a*sin (5) - cot(2*a)*tan(2*a) + a*cos (5)*cot(2*a)*tan(2*a)
----------------------------------------------------------------
                                   2                            
                         -1 + a*cos (5)
$$\frac{a \cos^{2}{\left(5 \right)} \tan{\left(2 a \right)} \cot{\left(2 a \right)} + a \sin^{2}{\left(5 \right)} - \tan{\left(2 a \right)} \cot{\left(2 a \right)} - 1}{a \cos^{2}{\left(5 \right)} - 1}$$ 
(-1 + a*sin(5)^2 - cot(2*a)*tan(2*a) + a*cos(5)^2*cot(2*a)*tan(2*a))/(-1 + a*cos(5)^2)
Abrimos la expresión [src] 
                                                                                                                            2                                                             8                                                                                    4                                                                                    2                                                                                    10                                                                                   6                                      
                                          1                                                      tan(a)                  cot (a)*tan(a)                                          640*a*sin (1)                                                                        200*a*sin (1)                                                                         25*a*sin (1)                                                                        256*a*sin  (1)                                                                       560*a*sin (1)                                   
- ---------------------------------------------------------------------------------- - ------------------------- + ------------------------- - ---------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------
                8               4              2               10               6           2                           2                                    8               4              2               10               6                    8               4              2               10               6                    8               4              2               10               6                    8               4              2               10               6                    8               4              2               10               6   
  -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)   - tan (a)*cot(a) + cot(a)   - tan (a)*cot(a) + cot(a)   -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)   -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)   -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)   -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)   -1 - 640*a*cos (1) - 200*a*cos (1) + 25*a*cos (1) + 256*a*cos  (1) + 560*a*cos (1)
$$- \frac{640 a \sin^{8}{\left(1 \right)}}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} - \frac{200 a \sin^{4}{\left(1 \right)}}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} + \frac{25 a \sin^{2}{\left(1 \right)}}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} + \frac{256 a \sin^{10}{\left(1 \right)}}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} + \frac{560 a \sin^{6}{\left(1 \right)}}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} - \frac{1}{- 200 a \cos^{4}{\left(1 \right)} - 640 a \cos^{8}{\left(1 \right)} + 256 a \cos^{10}{\left(1 \right)} + 25 a \cos^{2}{\left(1 \right)} + 560 a \cos^{6}{\left(1 \right)} - 1} + \frac{\tan{\left(a \right)} \cot^{2}{\left(a \right)}}{- \tan^{2}{\left(a \right)} \cot{\left(a \right)} + \cot{\left(a \right)}} - \frac{\tan{\left(a \right)}}{- \tan^{2}{\left(a \right)} \cot{\left(a \right)} + \cot{\left(a \right)}}$$ 
-1/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6) - tan(a)/(-tan(a)^2*cot(a) + cot(a)) + cot(a)^2*tan(a)/(-tan(a)^2*cot(a) + cot(a)) - 640*a*sin(1)^8/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6) - 200*a*sin(1)^4/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6) + 25*a*sin(1)^2/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6) + 256*a*sin(1)^10/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6) + 560*a*sin(1)^6/(-1 - 640*a*cos(1)^8 - 200*a*cos(1)^4 + 25*a*cos(1)^2 + 256*a*cos(1)^10 + 560*a*cos(1)^6)
Parte trigonométrica [src] 
      2 - a      
-----------------
          a      
1 - -------------
       2/     pi\
    csc |-5 + --|
        \     2 /
$$\frac{2 - a}{- \frac{a}{\csc^{2}{\left(-5 + \frac{\pi}{2} \right)}} + 1}$$ 
       2*(-2 + a)      
-----------------------
              /     pi\
-2 + a + a*sin|10 + --|
              \     2 /
$$\frac{2 \left(a - 2\right)}{a \sin{\left(\frac{\pi}{2} + 10 \right)} + a - 2}$$ 
    2 - a    
-------------
         2   
1 - a*cos (5)
$$\frac{2 - a}{- a \cos^{2}{\left(5 \right)} + 1}$$ 
        2*(-2 + a)       
-------------------------
           /        2   \
         a*\-1 + cot (5)/
-2 + a + ----------------
                  2      
           1 + cot (5)
$$\frac{2 \left(a - 2\right)}{\frac{a \left(-1 + \cot^{2}{\left(5 \right)}\right)}{\cot^{2}{\left(5 \right)} + 1} + a - 2}$$ 
           a    
    1 - ------- 
           2    
        csc (5) 
1 - ------------
            a   
    -1 + -------
            2   
         sec (5)
$$- \frac{- \frac{a}{\csc^{2}{\left(5 \right)}} + 1}{\frac{a}{\sec^{2}{\left(5 \right)}} - 1} + 1$$ 
         2 - a         
-----------------------
                      2
      /        2     \ 
    a*\-1 + cot (5/2)/ 
1 - -------------------
                     2 
      /       2     \  
      \1 + cot (5/2)/
$$\frac{2 - a}{- \frac{a \left(-1 + \cot^{2}{\left(\frac{5}{2} \right)}\right)^{2}}{\left(1 + \cot^{2}{\left(\frac{5}{2} \right)}\right)^{2}} + 1}$$ 
       2*(-2 + a)       
------------------------
           /       2   \
         a*\1 - tan (5)/
-2 + a + ---------------
                  2     
           1 + tan (5)
$$\frac{2 \left(a - 2\right)}{\frac{a \left(1 - \tan^{2}{\left(5 \right)}\right)}{1 + \tan^{2}{\left(5 \right)}} + a - 2}$$ 
                  2        
           4*a*tan (5/2)   
      1 - ---------------- 
                         2 
          /       2     \  
          \1 + tan (5/2)/  
1 - -----------------------
                          2
           /       2     \ 
         a*\1 - tan (5/2)/ 
    -1 + ------------------
                         2 
          /       2     \  
          \1 + tan (5/2)/
$$1 - \frac{- \frac{4 a \tan^{2}{\left(\frac{5}{2} \right)}}{\left(\tan^{2}{\left(\frac{5}{2} \right)} + 1\right)^{2}} + 1}{\frac{a \left(1 - \tan^{2}{\left(\frac{5}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{5}{2} \right)} + 1\right)^{2}} - 1}$$ 
      2*(-2 + a)      
----------------------
               a      
-2 + a + -------------
            /      pi\
         csc|-10 + --|
            \      2 /
$$\frac{2 \left(a - 2\right)}{\frac{a}{\csc{\left(-10 + \frac{\pi}{2} \right)}} + a - 2}$$ 
             2/    pi\
    1 - a*cos |5 - --|
              \    2 /
1 - ------------------
                2     
      -1 + a*cos (5)
$$1 - \frac{- a \cos^{2}{\left(5 - \frac{\pi}{2} \right)} + 1}{a \cos^{2}{\left(5 \right)} - 1}$$ 
      2 - a       
------------------
         2/    pi\
1 - a*sin |5 + --|
          \    2 /
$$\frac{2 - a}{- a \sin^{2}{\left(\frac{\pi}{2} + 5 \right)} + 1}$$ 
   2*(-2 + a)   
----------------
            a   
-2 + a + -------
         sec(10)
$$\frac{2 \left(a - 2\right)}{\frac{a}{\sec{\left(10 \right)}} + a - 2}$$ 
             2    
    1 - a*sin (5) 
1 - --------------
              2   
    -1 + a*cos (5)
$$- \frac{- a \sin^{2}{\left(5 \right)} + 1}{a \cos^{2}{\left(5 \right)} - 1} + 1$$ 
                  2         
           4*a*cot (5/2)    
      1 - ----------------  
                         2  
          /       2     \   
          \1 + cot (5/2)/   
1 - ------------------------
                           2
           /        2     \ 
         a*\-1 + cot (5/2)/ 
    -1 + -------------------
                          2 
           /       2     \  
           \1 + cot (5/2)/
$$1 - \frac{- \frac{4 a \cot^{2}{\left(\frac{5}{2} \right)}}{\left(1 + \cot^{2}{\left(\frac{5}{2} \right)}\right)^{2}} + 1}{\frac{a \left(-1 + \cot^{2}{\left(\frac{5}{2} \right)}\right)^{2}}{\left(1 + \cot^{2}{\left(\frac{5}{2} \right)}\right)^{2}} - 1}$$ 
        2 - a         
----------------------
                     2
      /       2     \ 
    a*\1 - tan (5/2)/ 
1 - ------------------
                    2 
     /       2     \  
     \1 + tan (5/2)/
$$\frac{2 - a}{- \frac{a \left(1 - \tan^{2}{\left(\frac{5}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{5}{2} \right)} + 1\right)^{2}} + 1}$$ 
     -4 + 2*a     
------------------
-2 + a + a*cos(10)
$$\frac{2 a - 4}{a \cos{\left(10 \right)} + a - 2}$$ 
              a       
       1 - -------    
              2       
           csc (5)    
1 - ------------------
               a      
    -1 + -------------
            2/     pi\
         csc |-5 + --|
             \     2 /
$$- \frac{- \frac{a}{\csc^{2}{\left(5 \right)}} + 1}{\frac{a}{\csc^{2}{\left(-5 + \frac{\pi}{2} \right)}} - 1} + 1$$ 
    2*(-2 + a)    
------------------
-2 + a + a*cos(10)
$$\frac{2 \left(a - 2\right)}{a \cos{\left(10 \right)} + a - 2}$$ 
   2 - a   
-----------
       a   
1 - -------
       2   
    sec (5)
$$\frac{2 - a}{- \frac{a}{\sec^{2}{\left(5 \right)}} + 1}$$ 
       2 - a       
-------------------
    a*(1 + cos(10))
1 - ---------------
           2
$$\frac{2 - a}{- \frac{a \left(\cos{\left(10 \right)} + 1\right)}{2} + 1}$$ 
             a      
    1 - ------------
           2/    pi\
        sec |5 - --|
            \    2 /
1 - ----------------
              a     
      -1 + -------  
              2     
           sec (5)
$$1 - \frac{- \frac{a}{\sec^{2}{\left(5 - \frac{\pi}{2} \right)}} + 1}{\frac{a}{\sec^{2}{\left(5 \right)}} - 1}$$ 
                2      
       1 - a*sin (5)   
1 - -------------------
              2/    pi\
    -1 + a*sin |5 + --|
               \    2 /
$$- \frac{- a \sin^{2}{\left(5 \right)} + 1}{a \sin^{2}{\left(\frac{\pi}{2} + 5 \right)} - 1} + 1$$ 
1 - (1 - a*sin(5)^2)/(-1 + a*sin(5 + pi/2)^2)
Respuesta numérica [src] 
cot(2*a)*tan(2*a) - (1.0 - 0.919535764538226*a)/(-1.0 + 0.0804642354617738*a)
cot(2*a)*tan(2*a) - (1.0 - 0.919535764538226*a)/(-1.0 + 0.0804642354617738*a)
Denominador común [src] 
          2                       
-1 + a*sin (5)                    
-------------- + cot(2*a)*tan(2*a)
          2                       
-1 + a*cos (5)
$$\frac{a \sin^{2}{\left(5 \right)} - 1}{a \cos^{2}{\left(5 \right)} - 1} + \tan{\left(2 a \right)} \cot{\left(2 a \right)}$$ 
(-1 + a*sin(5)^2)/(-1 + a*cos(5)^2) + cot(2*a)*tan(2*a)
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