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Simplificación de expresiones/ Descomposición en fracciones simples/ tan((pi/2)-x)^(2)-(1-sin(2*x))/(1+sin(2*x))

¿Cómo vas a descomponer esta tan((pi/2)-x)^(2)-(1-sin(2*x))/(1+sin(2*x)) expresión en fracciones?

Expresión a simplificar:

Solución

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