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Simplificación de expresiones/ Descomposición en fracciones simples/ tg(x-(\pi)/(4))-tg(x+(\pi)/(4))

¿Cómo vas a descomponer esta tg(x-(\pi)/(4))-tg(x+(\pi)/(4)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /    pi\      /    pi\
tan|x - --| - tan|x + --|
   \    4 /      \    4 /
$$\tan{\left(x - \frac{\pi}{4} \right)} - \tan{\left(x + \frac{\pi}{4} \right)}$$ 
tan(x - pi/4) - tan(x + pi/4)
Simplificación general [src] 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
-cot(x + pi/4) - tan(x + pi/4)
Respuesta numérica [src] 
-tan(x + pi/4) + tan(x - pi/4)
-tan(x + pi/4) + tan(x - pi/4)
Denominador racional [src] 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
-cot(x + pi/4) - tan(x + pi/4)
Denominador común [src] 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
-cot(x + pi/4) - tan(x + pi/4)
Unión de expresiones racionales [src] 
     /pi + 4*x\      /-pi + 4*x\
- tan|--------| + tan|---------|
     \   4    /      \    4    /
$$\tan{\left(\frac{4 x - \pi}{4} \right)} - \tan{\left(\frac{4 x + \pi}{4} \right)}$$ 
-tan((pi + 4*x)/4) + tan((-pi + 4*x)/4)
Combinatoria [src] 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
-cot(x + pi/4) - tan(x + pi/4)
Potencias [src] 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
  /     /    pi\      /     pi\\     /     /    pi\      /     pi\\
  |   I*|x - --|    I*|-x + --||     |   I*|x + --|    I*|-x - --||
  |     \    4 /      \     4 /|     |     \    4 /      \     4 /|
I*\- e           + e           /   I*\- e           + e           /
-------------------------------- - --------------------------------
      /    pi\      /     pi\            /    pi\      /     pi\   
    I*|x - --|    I*|-x + --|          I*|x + --|    I*|-x - --|   
      \    4 /      \     4 /            \    4 /      \     4 /   
   e           + e                    e           + e
$$- \frac{i \left(e^{i \left(- x - \frac{\pi}{4}\right)} - e^{i \left(x + \frac{\pi}{4}\right)}\right)}{e^{i \left(- x - \frac{\pi}{4}\right)} + e^{i \left(x + \frac{\pi}{4}\right)}} + \frac{i \left(e^{i \left(- x + \frac{\pi}{4}\right)} - e^{i \left(x - \frac{\pi}{4}\right)}\right)}{e^{i \left(- x + \frac{\pi}{4}\right)} + e^{i \left(x - \frac{\pi}{4}\right)}}$$ 
i*(-exp(i*(x - pi/4)) + exp(i*(-x + pi/4)))/(exp(i*(x - pi/4)) + exp(i*(-x + pi/4))) - i*(-exp(i*(x + pi/4)) + exp(i*(-x - pi/4)))/(exp(i*(x + pi/4)) + exp(i*(-x - pi/4)))
Abrimos la expresión [src] 
    1            1          tan(x)       cot(x)  
---------- - ---------- - ---------- - ----------
1 + cot(x)   1 - tan(x)   1 - tan(x)   1 + cot(x)
$$- \frac{\cot{\left(x \right)}}{\cot{\left(x \right)} + 1} + \frac{1}{\cot{\left(x \right)} + 1} - \frac{\tan{\left(x \right)}}{1 - \tan{\left(x \right)}} - \frac{1}{1 - \tan{\left(x \right)}}$$ 
1/(1 + cot(x)) - 1/(1 - tan(x)) - tan(x)/(1 - tan(x)) - cot(x)/(1 + cot(x))
Parte trigonométrica [src] 
     /     pi\      /    pi\ 
  csc|-x + --|   csc|x + --| 
     \     4 /      \    4 / 
- ------------ - ------------
     /    pi\       /     pi\
  csc|x + --|    csc|-x + --|
     \    4 /       \     4 /
$$- \frac{\csc{\left(- x + \frac{\pi}{4} \right)}}{\csc{\left(x + \frac{\pi}{4} \right)}} - \frac{\csc{\left(x + \frac{\pi}{4} \right)}}{\csc{\left(- x + \frac{\pi}{4} \right)}}$$ 
     /    pi\      /    pi\
  sec|x + --|   sec|x - --|
     \    4 /      \    4 /
- ----------- - -----------
     /    pi\      /    pi\
  sec|x - --|   sec|x + --|
     \    4 /      \    4 /
$$- \frac{\sec{\left(x - \frac{\pi}{4} \right)}}{\sec{\left(x + \frac{\pi}{4} \right)}} - \frac{\sec{\left(x + \frac{\pi}{4} \right)}}{\sec{\left(x - \frac{\pi}{4} \right)}}$$ 
      /pi      \
-2*csc|-- - 2*x|
      \2       /
$$- 2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}$$ 
     /    pi\      /    pi\
  cos|x + --|   sin|x + --|
     \    4 /      \    4 /
- ----------- - -----------
     /    pi\      /    pi\
  cos|x - --|   cos|x + --|
     \    4 /      \    4 /
$$- \frac{\sin{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}} - \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(x - \frac{\pi}{4} \right)}}$$ 
     -2      
-------------
   /pi      \
sin|-- + 2*x|
   \2       /
$$- \frac{2}{\sin{\left(2 x + \frac{\pi}{2} \right)}}$$ 
   /       2   \
-2*\1 + tan (x)/
----------------
         2      
  1 - tan (x)
$$- \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{1 - \tan^{2}{\left(x \right)}}$$ 
       2/    pi\        2/    pi\
  2*cos |x + --|   2*sin |x + --|
        \    4 /         \    4 /
- -------------- - --------------
     cos(2*x)         cos(2*x)
$$- \frac{2 \sin^{2}{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(2 x \right)}} - \frac{2 \cos^{2}{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(2 x \right)}}$$ 
       2/    pi\                 
  2*sin |x + --|                 
        \    4 /      cos(2*x)   
- -------------- - --------------
     cos(2*x)           2/    pi\
                   2*sin |x + --|
                         \    4 /
$$- \frac{2 \sin^{2}{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(2 x \right)}} - \frac{\cos{\left(2 x \right)}}{2 \sin^{2}{\left(x + \frac{\pi}{4} \right)}}$$ 
       1           /    pi\
- ----------- - tan|x + --|
     /    pi\      \    4 /
  tan|x + --|              
     \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \frac{1}{\tan{\left(x + \frac{\pi}{4} \right)}}$$ 
-2*sec(2*x)
$$- 2 \sec{\left(2 x \right)}$$ 
     /    pi\      /    pi\
- cot|x + --| - tan|x + --|
     \    4 /      \    4 /
$$- \tan{\left(x + \frac{\pi}{4} \right)} - \cot{\left(x + \frac{\pi}{4} \right)}$$ 
   /       2   \
-2*\1 + cot (x)/
----------------
          2     
  -1 + cot (x)
$$- \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{2}{\left(x \right)} - 1}$$ 
     /    pi\      /    pi\
  cos|x + --|   cos|x - --|
     \    4 /      \    4 /
- ----------- - -----------
     /    pi\      /    pi\
  cos|x - --|   cos|x + --|
     \    4 /      \    4 /
$$- \frac{\cos{\left(x - \frac{\pi}{4} \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}} - \frac{\cos{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(x - \frac{\pi}{4} \right)}}$$ 
   /    3*pi\      /    pi\
cos|x - ----|   cos|x - --|
   \     4  /      \    4 /
------------- - -----------
    /    pi\       /    pi\
 sin|x + --|    cos|x + --|
    \    4 /       \    4 /
$$- \frac{\cos{\left(x - \frac{\pi}{4} \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}} + \frac{\cos{\left(x - \frac{3 \pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{4} \right)}}$$ 
       1           /    pi\
- ----------- - cot|x + --|
     /    pi\      \    4 /
  cot|x + --|              
     \    4 /
$$- \cot{\left(x + \frac{\pi}{4} \right)} - \frac{1}{\cot{\left(x + \frac{\pi}{4} \right)}}$$ 
       2/    pi\        2/    3*pi\
  2*sin |x + --|   2*sin |x + ----|
        \    4 /         \     4  /
- -------------- - ----------------
     /pi      \        /pi      \  
  sin|-- + 2*x|     sin|-- + 2*x|  
     \2       /        \2       /
$$- \frac{2 \sin^{2}{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(2 x + \frac{\pi}{2} \right)}} - \frac{2 \sin^{2}{\left(x + \frac{3 \pi}{4} \right)}}{\sin{\left(2 x + \frac{\pi}{2} \right)}}$$ 
     /    pi\      /    pi\
  sec|x + --|   csc|x + --|
     \    4 /      \    4 /
- ----------- - -----------
     /    pi\      /    pi\
  csc|x + --|   sec|x + --|
     \    4 /      \    4 /
$$- \frac{\csc{\left(x + \frac{\pi}{4} \right)}}{\sec{\left(x + \frac{\pi}{4} \right)}} - \frac{\sec{\left(x + \frac{\pi}{4} \right)}}{\csc{\left(x + \frac{\pi}{4} \right)}}$$ 
  -2    
--------
cos(2*x)
$$- \frac{2}{\cos{\left(2 x \right)}}$$ 
       1             1     
- ----------- - -----------
     /    pi\      /    pi\
  cot|x + --|   tan|x + --|
     \    4 /      \    4 /
$$- \frac{1}{\cot{\left(x + \frac{\pi}{4} \right)}} - \frac{1}{\tan{\left(x + \frac{\pi}{4} \right)}}$$ 
-1/cot(x + pi/4) - 1/tan(x + pi/4)
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