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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
     /pi\   
  tan|--|   
     \8 /   
------------
       2/pi\
1 - tan |--|
        \8 /
$$\frac{\tan{\left(\frac{\pi}{8} \right)}}{1 - \tan^{2}{\left(\frac{\pi}{8} \right)}}$$ 
tan(pi/8)/(1 - tan(pi/8)^2)
Simplificación general [src] 
1/2
$$\frac{1}{2}$$ 
1/2
Respuesta numérica [src] 
0.500000000000000
0.500000000000000
Parte trigonométrica [src] 
               /-3*pi\            
            cos|-----|            
               \  8  /            
----------------------------------
/       2/-3*pi\\      ___________
|    cos |-----||     /       ___ 
|        \  8  /|    /  1   \/ 2  
|1 - -----------|*  /   - + ----- 
|           ___ | \/    2     4   
|     1   \/ 2  |                 
|     - + ----- |                 
\     2     4   /
$$\frac{\cos{\left(- \frac{3 \pi}{8} \right)}}{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{\cos^{2}{\left(- \frac{3 \pi}{8} \right)}}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 1\right)}$$ 
              1               
------------------------------
/      ___\ /         1      \
\1 + \/ 2 /*|1 - ------------|
            |               2|
            |    /      ___\ |
            \    \1 + \/ 2 / /
$$\frac{1}{\left(1 + \sqrt{2}\right) \left(1 - \frac{1}{\left(1 + \sqrt{2}\right)^{2}}\right)}$$ 
           ___   
    -1 + \/ 2    
-----------------
                2
    /       ___\ 
1 - \-1 + \/ 2 /
$$\frac{-1 + \sqrt{2}}{1 - \left(-1 + \sqrt{2}\right)^{2}}$$ 
      /      ___\ 
  ___ |    \/ 2 | 
\/ 2 *|1 - -----| 
      \      2  / 
------------------
                 2
      /      ___\ 
      |    \/ 2 | 
1 - 2*|1 - -----| 
      \      2  /
$$\frac{\sqrt{2} \left(1 - \frac{\sqrt{2}}{2}\right)}{1 - 2 \left(1 - \frac{\sqrt{2}}{2}\right)^{2}}$$ 
             ___________        
            /       ___         
           /  1   \/ 2          
          /   - - -----         
        \/    2     4           
--------------------------------
/          ___\                 
|    1   \/ 2 |      ___________
|    - - -----|     /       ___ 
|    2     4  |    /  1   \/ 2  
|1 - ---------|*  /   - + ----- 
|          ___| \/    2     4   
|    1   \/ 2 |                 
|    - + -----|                 
\    2     4  /
$$\frac{\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{\frac{1}{2} - \frac{\sqrt{2}}{4}}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 1\right)}$$ 
             ___________          
            /       ___           
          \/  2 - \/ 2            
----------------------------------
  /          ___\                 
  |    1   \/ 2 |      ___________
  |    - - -----|     /       ___ 
  |    2     4  |    /  1   \/ 2  
2*|1 - ---------|*  /   - + ----- 
  |          ___| \/    2     4   
  |    1   \/ 2 |                 
  |    - + -----|                 
  \    2     4  /
$$\frac{\sqrt{2 - \sqrt{2}}}{2 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \left(- \frac{\frac{1}{2} - \frac{\sqrt{2}}{4}}{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 1\right)}$$ 
sqrt(2 - sqrt(2))/(2*(1 - (1/2 - sqrt(2)/4)/(1/2 + sqrt(2)/4))*sqrt(1/2 + sqrt(2)/4))
Unión de expresiones racionales [src] 
           ___   
    -1 + \/ 2    
-----------------
                2
    /       ___\ 
1 - \-1 + \/ 2 /
$$\frac{-1 + \sqrt{2}}{1 - \left(-1 + \sqrt{2}\right)^{2}}$$ 
(-1 + sqrt(2))/(1 - (-1 + sqrt(2))^2)
Denominador racional [src] 
        ___ 
 -1 + \/ 2  
------------
       2/pi\
1 - tan |--|
        \8 /
$$\frac{-1 + \sqrt{2}}{1 - \tan^{2}{\left(\frac{\pi}{8} \right)}}$$ 
(-1 + sqrt(2))/(1 - tan(pi/8)^2)
Abrimos la expresión [src] 
                      ___    
       1            \/ 2     
- ------------ + ------------
           ___            ___
  -2 + 2*\/ 2    -2 + 2*\/ 2
$$- \frac{1}{-2 + 2 \sqrt{2}} + \frac{\sqrt{2}}{-2 + 2 \sqrt{2}}$$ 
-1/(-2 + 2*sqrt(2)) + sqrt(2)/(-2 + 2*sqrt(2))
Potencias [src] 
             /   pi*I    -pi*I \            
             |   ----    ------|            
             |    8        8   |            
           I*\- e     + e      /            
--------------------------------------------
/                       2\                  
|    /   pi*I    -pi*I \ |                  
|    |   ----    ------| | / -pi*I     pi*I\
|    |    8        8   | | | ------    ----|
|    \- e     + e      / | |   8        8  |
|1 + --------------------|*\e       + e    /
|                      2 |                  
|     / -pi*I     pi*I\  |                  
|     | ------    ----|  |                  
|     |   8        8  |  |                  
\     \e       + e    /  /
$$\frac{i \left(- e^{\frac{i \pi}{8}} + e^{- \frac{i \pi}{8}}\right)}{\left(1 + \frac{\left(- e^{\frac{i \pi}{8}} + e^{- \frac{i \pi}{8}}\right)^{2}}{\left(e^{- \frac{i \pi}{8}} + e^{\frac{i \pi}{8}}\right)^{2}}\right) \left(e^{- \frac{i \pi}{8}} + e^{\frac{i \pi}{8}}\right)}$$ 
           ___   
    -1 + \/ 2    
-----------------
                2
    /       ___\ 
1 - \-1 + \/ 2 /
$$\frac{-1 + \sqrt{2}}{1 - \left(-1 + \sqrt{2}\right)^{2}}$$ 
(-1 + sqrt(2))/(1 - (-1 + sqrt(2))^2)
Denominador común [src] 
1/2
$$\frac{1}{2}$$ 
1/2
Combinatoria [src] 
1/2
$$\frac{1}{2}$$ 
1/2
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