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Simplificación de expresiones/ Descomposición en fracciones simples/ tg2a*ctg2a-1-sin^2a/(cos^2a-1)

¿Cómo vas a descomponer esta tg2a*ctg2a-1-sin^2a/(cos^2a-1) expresión en fracciones?

Expresión a simplificar:

Solución

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