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Simplificación de expresiones/ Descomposición en fracciones simples/ tg(4*x)*ctg(4*x)-(1-sqrt(cos(9*x)))/(sqrt(sin(9*x))-1)

¿Cómo vas a descomponer esta tg(4*x)*ctg(4*x)-(1-sqrt(cos(9*x)))/(sqrt(sin(9*x))-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                          __________
                    1 - \/ cos(9*x) 
tan(4*x)*cot(4*x) - ----------------
                      __________    
                    \/ sin(9*x)  - 1
$$- \frac{1 - \sqrt{\cos{\left(9 x \right)}}}{\sqrt{\sin{\left(9 x \right)}} - 1} + \tan{\left(4 x \right)} \cot{\left(4 x \right)}$$ 
tan(4*x)*cot(4*x) - (1 - sqrt(cos(9*x)))/(sqrt(sin(9*x)) - 1)
Simplificación general [src] 
       __________     __________
-2 + \/ cos(9*x)  + \/ sin(9*x) 
--------------------------------
              __________        
       -1 + \/ sin(9*x)
$$\frac{\sqrt{\sin{\left(9 x \right)}} + \sqrt{\cos{\left(9 x \right)}} - 2}{\sqrt{\sin{\left(9 x \right)}} - 1}$$ 
(-2 + sqrt(cos(9*x)) + sqrt(sin(9*x)))/(-1 + sqrt(sin(9*x)))
Respuesta numérica [src] 
cot(4*x)*tan(4*x) - (1.0 - cos(9*x)^0.5)/(-1.0 + sin(9*x)^0.5)
cot(4*x)*tan(4*x) - (1.0 - cos(9*x)^0.5)/(-1.0 + sin(9*x)^0.5)
Potencias [src] 
       __________                    
-1 + \/ cos(9*x)                     
----------------- + cot(4*x)*tan(4*x)
       __________                    
-1 + \/ sin(9*x)
$$\tan{\left(4 x \right)} \cot{\left(4 x \right)} + \frac{\sqrt{\cos{\left(9 x \right)}} - 1}{\sqrt{\sin{\left(9 x \right)}} - 1}$$ 
                 __________________                                        
                /  -9*I*x    9*I*x                                         
               /  e         e                                              
        1 -   /   ------- + ------            /   4*I*x    -4*I*x\         
            \/       2        2             I*\- e      + e      /*cot(4*x)
- --------------------------------------- + -------------------------------
                _________________________            -4*I*x    4*I*x       
         ___   /    /   -9*I*x    9*I*x\            e       + e            
       \/ 2 *\/  -I*\- e       + e     /                                   
  -1 + ----------------------------------                                  
                       2
$$- \frac{1 - \sqrt{\frac{e^{9 i x}}{2} + \frac{e^{- 9 i x}}{2}}}{\frac{\sqrt{2} \sqrt{- i \left(e^{9 i x} - e^{- 9 i x}\right)}}{2} - 1} + \frac{i \left(- e^{4 i x} + e^{- 4 i x}\right) \cot{\left(4 x \right)}}{e^{4 i x} + e^{- 4 i x}}$$ 
-(1 - sqrt(exp(-9*i*x)/2 + exp(9*i*x)/2))/(-1 + sqrt(2)*sqrt(-i*(-exp(-9*i*x) + exp(9*i*x)))/2) + i*(-exp(4*i*x) + exp(-4*i*x))*cot(4*x)/(exp(-4*i*x) + exp(4*i*x))
Denominador común [src] 
       __________                    
-1 + \/ cos(9*x)                     
----------------- + cot(4*x)*tan(4*x)
       __________                    
-1 + \/ sin(9*x)
$$\tan{\left(4 x \right)} \cot{\left(4 x \right)} + \frac{\sqrt{\cos{\left(9 x \right)}} - 1}{\sqrt{\sin{\left(9 x \right)}} - 1}$$ 
(-1 + sqrt(cos(9*x)))/(-1 + sqrt(sin(9*x))) + cot(4*x)*tan(4*x)
Denominador racional [src] 
       __________     __________     __________   __________                                                 
-1 + \/ cos(9*x)  - \/ sin(9*x)  + \/ cos(9*x) *\/ sin(9*x)  - cot(4*x)*tan(4*x) + cot(4*x)*sin(9*x)*tan(4*x)
-------------------------------------------------------------------------------------------------------------
                                                -1 + sin(9*x)
$$\frac{\sqrt{\sin{\left(9 x \right)}} \sqrt{\cos{\left(9 x \right)}} - \sqrt{\sin{\left(9 x \right)}} + \sin{\left(9 x \right)} \tan{\left(4 x \right)} \cot{\left(4 x \right)} + \sqrt{\cos{\left(9 x \right)}} - \tan{\left(4 x \right)} \cot{\left(4 x \right)} - 1}{\sin{\left(9 x \right)} - 1}$$ 
(-1 + sqrt(cos(9*x)) - sqrt(sin(9*x)) + sqrt(cos(9*x))*sqrt(sin(9*x)) - cot(4*x)*tan(4*x) + cot(4*x)*sin(9*x)*tan(4*x))/(-1 + sin(9*x))
Unión de expresiones racionales [src] 
       __________   /       __________\                  
-1 + \/ cos(9*x)  + \-1 + \/ sin(9*x) /*cot(4*x)*tan(4*x)
---------------------------------------------------------
                           __________                    
                    -1 + \/ sin(9*x)
$$\frac{\left(\sqrt{\sin{\left(9 x \right)}} - 1\right) \tan{\left(4 x \right)} \cot{\left(4 x \right)} + \sqrt{\cos{\left(9 x \right)}} - 1}{\sqrt{\sin{\left(9 x \right)}} - 1}$$ 
(-1 + sqrt(cos(9*x)) + (-1 + sqrt(sin(9*x)))*cot(4*x)*tan(4*x))/(-1 + sqrt(sin(9*x)))
Abrimos la expresión [src] 
                                                                                      ____________________________________________________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
                                                                                     /          7             3                        9             5                                                             3                                                                                                                                                                                                            2                                                                                                      4       3                                                                                                4                                                                                                       2       3                                            
                                       1                                           \/  - 576*cos (x) - 120*cos (x) + 9*cos(x) + 256*cos (x) + 432*cos (x)                                                     4*tan (x)                                                                                                4*tan(x)                                                                                           24*cot (x)*tan(x)                                                                                       4*cot (x)*tan (x)                                                                                        4*cot (x)*tan(x)                                                                                       24*cot (x)*tan (x)                                         
- ---------------------------------------------------------------------------- + ---------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------
          ____________________________________________________________________           ____________________________________________________________________                    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          
         /          7             3                        9             5              /          7             3                        9             5       -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)   -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)   -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)   -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)   -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)   -4*cot(x) + 4*cot (x) - 24*cot (x)*tan (x) - 4*tan (x)*cot(x) + 4*cot (x)*tan (x) + 24*tan (x)*cot(x)
  -1 + \/  - 576*sin (x) - 120*sin (x) + 9*sin(x) + 256*sin (x) + 432*sin (x)    -1 + \/  - 576*sin (x) - 120*sin (x) + 9*sin(x) + 256*sin (x) + 432*sin (x)
$$- \frac{4 \tan^{3}{\left(x \right)} \cot^{4}{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} + \frac{24 \tan^{3}{\left(x \right)} \cot^{2}{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} - \frac{4 \tan^{3}{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} + \frac{4 \tan{\left(x \right)} \cot^{4}{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} - \frac{24 \tan{\left(x \right)} \cot^{2}{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} + \frac{4 \tan{\left(x \right)}}{4 \tan^{4}{\left(x \right)} \cot^{3}{\left(x \right)} - 4 \tan^{4}{\left(x \right)} \cot{\left(x \right)} - 24 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \cot{\left(x \right)} + 4 \cot^{3}{\left(x \right)} - 4 \cot{\left(x \right)}} + \frac{\sqrt{256 \cos^{9}{\left(x \right)} - 576 \cos^{7}{\left(x \right)} + 432 \cos^{5}{\left(x \right)} - 120 \cos^{3}{\left(x \right)} + 9 \cos{\left(x \right)}}}{\sqrt{256 \sin^{9}{\left(x \right)} - 576 \sin^{7}{\left(x \right)} + 432 \sin^{5}{\left(x \right)} - 120 \sin^{3}{\left(x \right)} + 9 \sin{\left(x \right)}} - 1} - \frac{1}{\sqrt{256 \sin^{9}{\left(x \right)} - 576 \sin^{7}{\left(x \right)} + 432 \sin^{5}{\left(x \right)} - 120 \sin^{3}{\left(x \right)} + 9 \sin{\left(x \right)}} - 1}$$ 
-1/(-1 + sqrt(-576*sin(x)^7 - 120*sin(x)^3 + 9*sin(x) + 256*sin(x)^9 + 432*sin(x)^5)) + sqrt(-576*cos(x)^7 - 120*cos(x)^3 + 9*cos(x) + 256*cos(x)^9 + 432*cos(x)^5)/(-1 + sqrt(-576*sin(x)^7 - 120*sin(x)^3 + 9*sin(x) + 256*sin(x)^9 + 432*sin(x)^5)) - 4*tan(x)^3/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x)) + 4*tan(x)/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x)) - 24*cot(x)^2*tan(x)/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x)) - 4*cot(x)^4*tan(x)^3/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x)) + 4*cot(x)^4*tan(x)/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x)) + 24*cot(x)^2*tan(x)^3/(-4*cot(x) + 4*cot(x)^3 - 24*cot(x)^3*tan(x)^2 - 4*tan(x)^4*cot(x) + 4*cot(x)^3*tan(x)^4 + 24*tan(x)^2*cot(x))
Combinatoria [src] 
       __________                         __________                  
-1 + \/ cos(9*x)  - cot(4*x)*tan(4*x) + \/ sin(9*x) *cot(4*x)*tan(4*x)
----------------------------------------------------------------------
                                 __________                           
                          -1 + \/ sin(9*x)
$$\frac{\sqrt{\sin{\left(9 x \right)}} \tan{\left(4 x \right)} \cot{\left(4 x \right)} + \sqrt{\cos{\left(9 x \right)}} - \tan{\left(4 x \right)} \cot{\left(4 x \right)} - 1}{\sqrt{\sin{\left(9 x \right)}} - 1}$$ 
(-1 + sqrt(cos(9*x)) - cot(4*x)*tan(4*x) + sqrt(sin(9*x))*cot(4*x)*tan(4*x))/(-1 + sqrt(sin(9*x)))
Parte trigonométrica [src] 
                   ________________   
                  /         2/9*x\    
                 /  -1 + cot |---|    
                /            \ 2 /    
       1 -     /    --------------    
              /            2/9*x\     
             /      1 + cot |---|     
           \/               \ 2 /     
1 - ----------------------------------
                       _______________
                      /       /9*x\   
                     /     cot|---|   
           ___      /         \ 2 /   
    -1 + \/ 2 *    /    ------------- 
                  /            2/9*x\ 
                 /      1 + cot |---| 
               \/               \ 2 /
$$- \frac{1 - \sqrt{\frac{\cot^{2}{\left(\frac{9 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{9 x}{2} \right)} + 1}}}{\sqrt{2} \sqrt{\frac{\cot{\left(\frac{9 x}{2} \right)}}{\cot^{2}{\left(\frac{9 x}{2} \right)} + 1}} - 1} + 1$$ 
                   _______________    
                  /        2/9*x\     
                 /  1 - tan |---|     
                /           \ 2 /     
       1 -     /    -------------     
              /            2/9*x\     
             /      1 + tan |---|     
           \/               \ 2 /     
1 - ----------------------------------
                       _______________
                      /       /9*x\   
                     /     tan|---|   
           ___      /         \ 2 /   
    -1 + \/ 2 *    /    ------------- 
                  /            2/9*x\ 
                 /      1 + tan |---| 
               \/               \ 2 /
$$- \frac{1 - \sqrt{\frac{1 - \tan^{2}{\left(\frac{9 x}{2} \right)}}{\tan^{2}{\left(\frac{9 x}{2} \right)} + 1}}}{\sqrt{2} \sqrt{\frac{\tan{\left(\frac{9 x}{2} \right)}}{\tan^{2}{\left(\frac{9 x}{2} \right)} + 1}} - 1} + 1$$ 
           __________
     1 - \/ cos(9*x) 
1 - -----------------
           __________
    -1 + \/ sin(9*x)
$$- \frac{1 - \sqrt{\cos{\left(9 x \right)}}}{\sqrt{\sin{\left(9 x \right)}} - 1} + 1$$ 
              _______________
             /       1       
    1 -     /  ------------- 
           /      /pi      \ 
          /    csc|-- - 9*x| 
        \/        \2       / 
1 - -------------------------
                __________   
               /    1        
       -1 +   /  --------    
            \/   csc(9*x)
$$- \frac{1 - \sqrt{\frac{1}{\csc{\left(- 9 x + \frac{\pi}{2} \right)}}}}{\sqrt{\frac{1}{\csc{\left(9 x \right)}}} - 1} + 1$$ 
                __________    
               /    1         
        1 -   /  --------     
            \/   sec(9*x)     
1 - --------------------------
               _______________
              /       1       
    -1 +     /  ------------- 
            /      /      pi\ 
           /    sec|9*x - --| 
         \/        \      2 /
$$- \frac{1 - \sqrt{\frac{1}{\sec{\left(9 x \right)}}}}{\sqrt{\frac{1}{\sec{\left(9 x - \frac{\pi}{2} \right)}}} - 1} + 1$$ 
              __________    
        1 - \/ cos(9*x)     
1 - ------------------------
             _______________
            /    /      pi\ 
    -1 +   /  cos|9*x - --| 
         \/      \      2 /
$$- \frac{1 - \sqrt{\cos{\left(9 x \right)}}}{\sqrt{\cos{\left(9 x - \frac{\pi}{2} \right)}} - 1} + 1$$ 
            _______________
           /    /pi      \ 
    1 -   /  sin|-- + 9*x| 
        \/      \2       / 
1 - -----------------------
              __________   
       -1 + \/ sin(9*x)
$$- \frac{1 - \sqrt{\sin{\left(9 x + \frac{\pi}{2} \right)}}}{\sqrt{\sin{\left(9 x \right)}} - 1} + 1$$ 
          __________
    1 - \/ cos(9*x) 
1 - ----------------
      __________    
    \/ sin(9*x)  - 1
$$- \frac{1 - \sqrt{\cos{\left(9 x \right)}}}{\sqrt{\sin{\left(9 x \right)}} - 1} + 1$$ 
             __________
            /    1     
     1 -   /  -------- 
         \/   sec(9*x) 
1 - -------------------
             __________
            /    1     
    -1 +   /  -------- 
         \/   csc(9*x)
$$- \frac{1 - \sqrt{\frac{1}{\sec{\left(9 x \right)}}}}{\sqrt{\frac{1}{\csc{\left(9 x \right)}}} - 1} + 1$$ 
1 - (1 - sqrt(1/sec(9*x)))/(-1 + sqrt(1/csc(9*x)))
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