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Simplificación de expresiones/ Descomposición en fracciones simples/ tg^2(x)*ctg^2(x)-(1-sin^2(5x))/(cos^2(5x))-1

¿Cómo vas a descomponer esta tg^2(x)*ctg^2(x)-(1-sin^2(5x))/(cos^2(5x))-1 expresión en fracciones?

Expresión a simplificar:

Solución

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