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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
      /pi\    
   tan|--|    
      \8 /    
--------------
         2/pi\
2 - 2*tan |--|
          \8 /
$$\frac{\tan{\left(\frac{\pi}{8} \right)}}{2 - 2 \tan^{2}{\left(\frac{\pi}{8} \right)}}$$ 
tan(pi/8)/(2 - 2*tan(pi/8)^2)
Simplificación general [src] 
1/4
$$\frac{1}{4}$$ 
1/4
Respuesta numérica [src] 
0.250000000000000
0.250000000000000
Potencias [src] 
              /   pi*I    -pi*I \             
              |   ----    ------|             
              |    8        8   |             
            I*\- e     + e      /             
----------------------------------------------
/                         2\                  
|      /   pi*I    -pi*I \ |                  
|      |   ----    ------| | / -pi*I     pi*I\
|      |    8        8   | | | ------    ----|
|    2*\- e     + e      / | |   8        8  |
|2 + ----------------------|*\e       + e    /
|                       2  |                  
|      / -pi*I     pi*I\   |                  
|      | ------    ----|   |                  
|      |   8        8  |   |                  
\      \e       + e    /   /
$$\frac{i \left(- e^{\frac{i \pi}{8}} + e^{- \frac{i \pi}{8}}\right)}{\left(2 + \frac{2 \left(- e^{\frac{i \pi}{8}} + e^{- \frac{i \pi}{8}}\right)^{2}}{\left(e^{- \frac{i \pi}{8}} + e^{\frac{i \pi}{8}}\right)^{2}}\right) \left(e^{- \frac{i \pi}{8}} + e^{\frac{i \pi}{8}}\right)}$$ 
            ___    
     -1 + \/ 2     
-------------------
                  2
      /       ___\ 
2 - 2*\-1 + \/ 2 /
$$\frac{-1 + \sqrt{2}}{2 - 2 \left(-1 + \sqrt{2}\right)^{2}}$$ 
(-1 + sqrt(2))/(2 - 2*(-1 + sqrt(2))^2)
Denominador racional [src] 
         ___  
  -1 + \/ 2   
--------------
         2/pi\
2 - 2*tan |--|
          \8 /
$$\frac{-1 + \sqrt{2}}{2 - 2 \tan^{2}{\left(\frac{\pi}{8} \right)}}$$ 
(-1 + sqrt(2))/(2 - 2*tan(pi/8)^2)
Denominador común [src] 
1/4
$$\frac{1}{4}$$ 
1/4
Combinatoria [src] 
1/4
$$\frac{1}{4}$$ 
1/4
Unión de expresiones racionales [src] 
             ___     
      -1 + \/ 2      
---------------------
  /                2\
  |    /       ___\ |
2*\1 - \-1 + \/ 2 / /
$$\frac{-1 + \sqrt{2}}{2 \left(1 - \left(-1 + \sqrt{2}\right)^{2}\right)}$$ 
(-1 + sqrt(2))/(2*(1 - (-1 + sqrt(2))^2))
Abrimos la expresión [src] 
                      ___    
       1            \/ 2     
- ------------ + ------------
           ___            ___
  -4 + 4*\/ 2    -4 + 4*\/ 2
$$- \frac{1}{-4 + 4 \sqrt{2}} + \frac{\sqrt{2}}{-4 + 4 \sqrt{2}}$$ 
-1/(-4 + 4*sqrt(2)) + sqrt(2)/(-4 + 4*sqrt(2))
Parte trigonométrica [src] 
      /      ___\ 
  ___ |    \/ 2 | 
\/ 2 *|1 - -----| 
      \      2  / 
------------------
                 2
      /      ___\ 
      |    \/ 2 | 
2 - 4*|1 - -----| 
      \      2  /
$$\frac{\sqrt{2} \left(1 - \frac{\sqrt{2}}{2}\right)}{2 - 4 \left(1 - \frac{\sqrt{2}}{2}\right)^{2}}$$ 
              1               
------------------------------
/      ___\ /         2      \
\1 + \/ 2 /*|2 - ------------|
            |               2|
            |    /      ___\ |
            \    \1 + \/ 2 / /
$$\frac{1}{\left(1 + \sqrt{2}\right) \left(2 - \frac{2}{\left(1 + \sqrt{2}\right)^{2}}\right)}$$ 
               ___________            
              /       ___             
            \/  2 - \/ 2              
--------------------------------------
                   /      /      ___\\
       ___________ |      |1   \/ 2 ||
      /       ___  |    2*|- - -----||
     /  1   \/ 2   |      \2     4  /|
2*  /   - + ----- *|2 - -------------|
  \/    2     4    |            ___  |
                   |      1   \/ 2   |
                   |      - + -----  |
                   \      2     4    /
$$\frac{\sqrt{2 - \sqrt{2}}}{2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{2 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2\right)}$$ 
                /-3*pi\             
             cos|-----|             
                \  8  /             
------------------------------------
     ___________ /         2/-3*pi\\
    /       ___  |    2*cos |-----||
   /  1   \/ 2   |          \  8  /|
  /   - + ----- *|2 - -------------|
\/    2     4    |            ___  |
                 |      1   \/ 2   |
                 |      - + -----  |
                 \      2     4    /
$$\frac{\cos{\left(- \frac{3 \pi}{8} \right)}}{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{2 \cos^{2}{\left(- \frac{3 \pi}{8} \right)}}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2\right)}$$ 
            ___    
     -1 + \/ 2     
-------------------
                  2
      /       ___\ 
2 - 2*\-1 + \/ 2 /
$$\frac{-1 + \sqrt{2}}{2 - 2 \left(-1 + \sqrt{2}\right)^{2}}$$ 
               ___________          
              /       ___           
             /  1   \/ 2            
            /   - - -----           
          \/    2     4             
------------------------------------
                 /      /      ___\\
     ___________ |      |1   \/ 2 ||
    /       ___  |    2*|- - -----||
   /  1   \/ 2   |      \2     4  /|
  /   - + ----- *|2 - -------------|
\/    2     4    |            ___  |
                 |      1   \/ 2   |
                 |      - + -----  |
                 \      2     4    /
$$\frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{2 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 2\right)}$$ 
sqrt(1/2 - sqrt(2)/4)/(sqrt(1/2 + sqrt(2)/4)*(2 - 2*(1/2 - sqrt(2)/4)/(1/2 + sqrt(2)/4)))
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