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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(x+pi/8)/2-1/(2*tan(x+pi/8))

¿Cómo vas a descomponer esta tan(x+pi/8)/2-1/(2*tan(x+pi/8)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /    pi\                
tan|x + --|                
   \    8 /         1      
----------- - -------------
     2             /    pi\
              2*tan|x + --|
                   \    8 /
$$\frac{\tan{\left(x + \frac{\pi}{8} \right)}}{2} - \frac{1}{2 \tan{\left(x + \frac{\pi}{8} \right)}}$$ 
tan(x + pi/8)/2 - 1/(2*tan(x + pi/8))
Descomposición de una fracción [src] 
tan(x + pi/8)/2 - 1/(2*tan(x + pi/8))
$$\frac{\tan{\left(x + \frac{\pi}{8} \right)}}{2} - \frac{1}{2 \tan{\left(x + \frac{\pi}{8} \right)}}$$ 
   /    pi\                
tan|x + --|                
   \    8 /         1      
----------- - -------------
     2             /    pi\
              2*tan|x + --|
                   \    8 /
Simplificación general [src] 
        2/    pi\
-1 + tan |x + --|
         \    8 /
-----------------
       /    pi\  
  2*tan|x + --|  
       \    8 /
$$\frac{\tan^{2}{\left(x + \frac{\pi}{8} \right)} - 1}{2 \tan{\left(x + \frac{\pi}{8} \right)}}$$ 
(-1 + tan(x + pi/8)^2)/(2*tan(x + pi/8))
Respuesta numérica [src] 
0.5*tan(x + pi/8) - 0.5/tan(x + pi/8)
0.5*tan(x + pi/8) - 0.5/tan(x + pi/8)
Denominador racional [src] 
          2/    pi\
-2 + 2*tan |x + --|
           \    8 /
-------------------
        /    pi\   
   4*tan|x + --|   
        \    8 /
$$\frac{2 \tan^{2}{\left(x + \frac{\pi}{8} \right)} - 2}{4 \tan{\left(x + \frac{\pi}{8} \right)}}$$ 
(-2 + 2*tan(x + pi/8)^2)/(4*tan(x + pi/8))
Unión de expresiones racionales [src] 
        2/pi + 8*x\
-1 + tan |--------|
         \   8    /
-------------------
       /pi + 8*x\  
  2*tan|--------|  
       \   8    /
$$\frac{\tan^{2}{\left(\frac{8 x + \pi}{8} \right)} - 1}{2 \tan{\left(\frac{8 x + \pi}{8} \right)}}$$ 
(-1 + tan((pi + 8*x)/8)^2)/(2*tan((pi + 8*x)/8))
Combinatoria [src] 
/       /    pi\\ /        /    pi\\
|1 + tan|x + --||*|-1 + tan|x + --||
\       \    8 // \        \    8 //
------------------------------------
                /    pi\            
           2*tan|x + --|            
                \    8 /
$$\frac{\left(\tan{\left(x + \frac{\pi}{8} \right)} - 1\right) \left(\tan{\left(x + \frac{\pi}{8} \right)} + 1\right)}{2 \tan{\left(x + \frac{\pi}{8} \right)}}$$ 
(1 + tan(x + pi/8))*(-1 + tan(x + pi/8))/(2*tan(x + pi/8))
Potencias [src] 
   /   /    pi\      /     pi\\      /     /    pi\      /     pi\\
   | I*|x + --|    I*|-x - --||      |   I*|x + --|    I*|-x - --||
   |   \    8 /      \     8 /|      |     \    8 /      \     8 /|
 I*\e           + e           /    I*\- e           + e           /
-------------------------------- + --------------------------------
  /     /    pi\      /     pi\\      /   /    pi\      /     pi\\ 
  |   I*|x + --|    I*|-x - --||      | I*|x + --|    I*|-x - --|| 
  |     \    8 /      \     8 /|      |   \    8 /      \     8 /| 
2*\- e           + e           /    2*\e           + e           /
$$\frac{i \left(e^{i \left(- x - \frac{\pi}{8}\right)} - e^{i \left(x + \frac{\pi}{8}\right)}\right)}{2 \left(e^{i \left(- x - \frac{\pi}{8}\right)} + e^{i \left(x + \frac{\pi}{8}\right)}\right)} + \frac{i \left(e^{i \left(- x - \frac{\pi}{8}\right)} + e^{i \left(x + \frac{\pi}{8}\right)}\right)}{2 \left(e^{i \left(- x - \frac{\pi}{8}\right)} - e^{i \left(x + \frac{\pi}{8}\right)}\right)}$$ 
i*(exp(i*(x + pi/8)) + exp(i*(-x - pi/8)))/(2*(-exp(i*(x + pi/8)) + exp(i*(-x - pi/8)))) + i*(-exp(i*(x + pi/8)) + exp(i*(-x - pi/8)))/(2*(exp(i*(x + pi/8)) + exp(i*(-x - pi/8))))
Abrimos la expresión [src] 
                                                                          ___                                                                                 ___            
                1                            1                          \/ 2                            tan(x)                       tan(x)                 \/ 2 *tan(x)     
- ----------------------------- - ----------------------- + ----------------------------- + ----------------------------- - ----------------------- + -----------------------
    /      ___                \     /       ___         \     /      ___                \     /      ___                \     /       ___         \     /       ___         \
  2*\1 - \/ 2 *tan(x) + tan(x)/   2*\-1 + \/ 2  + tan(x)/   2*\1 - \/ 2 *tan(x) + tan(x)/   2*\1 - \/ 2 *tan(x) + tan(x)/   2*\-1 + \/ 2  + tan(x)/   2*\-1 + \/ 2  + tan(x)/
$$- \frac{\tan{\left(x \right)}}{2 \left(\tan{\left(x \right)} - 1 + \sqrt{2}\right)} + \frac{\sqrt{2} \tan{\left(x \right)}}{2 \left(\tan{\left(x \right)} - 1 + \sqrt{2}\right)} - \frac{1}{2 \left(\tan{\left(x \right)} - 1 + \sqrt{2}\right)} + \frac{\tan{\left(x \right)}}{2 \left(- \sqrt{2} \tan{\left(x \right)} + \tan{\left(x \right)} + 1\right)} - \frac{1}{2 \left(- \sqrt{2} \tan{\left(x \right)} + \tan{\left(x \right)} + 1\right)} + \frac{\sqrt{2}}{2 \left(- \sqrt{2} \tan{\left(x \right)} + \tan{\left(x \right)} + 1\right)}$$ 
-1/(2*(1 - sqrt(2)*tan(x) + tan(x))) - 1/(2*(-1 + sqrt(2) + tan(x))) + sqrt(2)/(2*(1 - sqrt(2)*tan(x) + tan(x))) + tan(x)/(2*(1 - sqrt(2)*tan(x) + tan(x))) - tan(x)/(2*(-1 + sqrt(2) + tan(x))) + sqrt(2)*tan(x)/(2*(-1 + sqrt(2) + tan(x)))
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