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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(pi/4-a)/(1-sin(2a))^(-1)-(sin(pi/4-a))/(1+sin(2a))^(-1)

¿Cómo vas a descomponer esta cos(pi/4-a)/(1-sin(2a))^(-1)-(sin(pi/4-a))/(1+sin(2a))^(-1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /pi    \         /pi    \  
 cos|-- - a|      sin|-- - a|  
    \4     /         \4     /  
-------------- - --------------
/     1      \   /     1      \
|------------|   |------------|
\1 - sin(2*a)/   \1 + sin(2*a)/
−sin⁡(−a+π4)1sin⁡(2a)+1+cos⁡(−a+π4)11−sin⁡(2a)- \frac{\sin{\left(- a + \frac{\pi}{4} \right)}}{\frac{1}{\sin{\left(2 a \right)} + 1}} + \frac{\cos{\left(- a + \frac{\pi}{4} \right)}}{\frac{1}{1 - \sin{\left(2 a \right)}}} 
cos(pi/4 - a)/1/(1 - sin(2*a)) - sin(pi/4 - a)/1/(1 + sin(2*a))
Simplificación general [src] 
  ___                     
\/ 2 *(-sin(3*a) + sin(a))
--------------------------
            2
2(sin⁡(a)−sin⁡(3a))2\frac{\sqrt{2} \left(\sin{\left(a \right)} - \sin{\left(3 a \right)}\right)}{2} 
sqrt(2)*(-sin(3*a) + sin(a))/2
Respuesta numérica [src] 
(1.0 - sin(2*a))*cos(pi/4 - a) - (1.0 + sin(2*a))*sin(pi/4 - a)
(1.0 - sin(2*a))*cos(pi/4 - a) - (1.0 + sin(2*a))*sin(pi/4 - a)
Denominador común [src] 
     /    pi\      /    pi\                        /    pi\      /    pi\
- cos|a + --| - cos|a + --|*sin(2*a) - sin(2*a)*sin|a + --| + sin|a + --|
     \    4 /      \    4 /                        \    4 /      \    4 /
−sin⁡(2a)sin⁡(a+π4)−sin⁡(2a)cos⁡(a+π4)+sin⁡(a+π4)−cos⁡(a+π4)- \sin{\left(2 a \right)} \sin{\left(a + \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a + \frac{\pi}{4} \right)} + \sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)} 
-cos(a + pi/4) - cos(a + pi/4)*sin(2*a) - sin(2*a)*sin(a + pi/4) + sin(a + pi/4)
Combinatoria [src] 
     /    pi\      /    pi\                        /    pi\      /    pi\
- cos|a + --| - cos|a + --|*sin(2*a) - sin(2*a)*sin|a + --| + sin|a + --|
     \    4 /      \    4 /                        \    4 /      \    4 /
−sin⁡(2a)sin⁡(a+π4)−sin⁡(2a)cos⁡(a+π4)+sin⁡(a+π4)−cos⁡(a+π4)- \sin{\left(2 a \right)} \sin{\left(a + \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a + \frac{\pi}{4} \right)} + \sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)} 
-cos(a + pi/4) - cos(a + pi/4)*sin(2*a) - sin(2*a)*sin(a + pi/4) + sin(a + pi/4)
Denominador racional [src] 
            /    pi\      /    pi\               /    pi\      /    pi\
sin(2*a)*sin|a - --| - cos|a - --|*sin(2*a) + cos|a - --| + sin|a - --|
            \    4 /      \    4 /               \    4 /      \    4 /
sin⁡(2a)sin⁡(a−π4)−sin⁡(2a)cos⁡(a−π4)+sin⁡(a−π4)+cos⁡(a−π4)\sin{\left(2 a \right)} \sin{\left(a - \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a - \frac{\pi}{4} \right)} + \sin{\left(a - \frac{\pi}{4} \right)} + \cos{\left(a - \frac{\pi}{4} \right)} 
sin(2*a)*sin(a - pi/4) - cos(a - pi/4)*sin(2*a) + cos(a - pi/4) + sin(a - pi/4)
Unión de expresiones racionales [src] 
                  /pi - 4*a\                     /pi - 4*a\
(1 - sin(2*a))*cos|--------| - (1 + sin(2*a))*sin|--------|
                  \   4    /                     \   4    /
(1−sin⁡(2a))cos⁡(π−4a4)−(sin⁡(2a)+1)sin⁡(π−4a4)\left(1 - \sin{\left(2 a \right)}\right) \cos{\left(\frac{\pi - 4 a}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \sin{\left(\frac{\pi - 4 a}{4} \right)} 
(1 - sin(2*a))*cos((pi - 4*a)/4) - (1 + sin(2*a))*sin((pi - 4*a)/4)
Potencias [src] 
                                                                                           /     /    pi\      /     pi\\
                             /   /    pi\      /     pi\\     /      /   -2*I*a    2*I*a\\ |   I*|a - --|    I*|-a + --||
                             | I*|a - --|    I*|-a + --||     |    I*\- e       + e     /| |     \    4 /      \     4 /|
/      /   -2*I*a    2*I*a\\ |   \    4 /      \     4 /|   I*|1 - ----------------------|*\- e           + e           /
|    I*\- e       + e     /| |e             e           |     \              2           /                               
|1 + ----------------------|*|----------- + ------------| + -------------------------------------------------------------
\              2           / \     2             2      /                                 2
i(−i(e2ia−e−2ia)2+1)(ei(−a+π4)−ei(a−π4))2+(i(e2ia−e−2ia)2+1)(ei(−a+π4)2+ei(a−π4)2)\frac{i \left(- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + 1\right) \left(e^{i \left(- a + \frac{\pi}{4}\right)} - e^{i \left(a - \frac{\pi}{4}\right)}\right)}{2} + \left(\frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + 1\right) \left(\frac{e^{i \left(- a + \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(a - \frac{\pi}{4}\right)}}{2}\right) 
                  /    pi\                     /    pi\
(1 - sin(2*a))*sin|a + --| - (1 + sin(2*a))*cos|a + --|
                  \    4 /                     \    4 /
(1−sin⁡(2a))sin⁡(a+π4)−(sin⁡(2a)+1)cos⁡(a+π4)\left(1 - \sin{\left(2 a \right)}\right) \sin{\left(a + \frac{\pi}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \cos{\left(a + \frac{\pi}{4} \right)} 
(1 - sin(2*a))*sin(a + pi/4) - (1 + sin(2*a))*cos(a + pi/4)
Abrimos la expresión [src] 
  ___              ___    2          
\/ 2 *sin(a) - 2*\/ 2 *cos (a)*sin(a)
−22sin⁡(a)cos⁡2(a)+2sin⁡(a)- 2 \sqrt{2} \sin{\left(a \right)} \cos^{2}{\left(a \right)} + \sqrt{2} \sin{\left(a \right)} 
sqrt(2)*sin(a) - 2*sqrt(2)*cos(a)^2*sin(a)
Compilar la expresión [src] 
                  /pi    \                     /pi    \
(1 - sin(2*a))*cos|-- - a| - (1 + sin(2*a))*sin|-- - a|
                  \4     /                     \4     /
(1−sin⁡(2a))cos⁡(−a+π4)−(sin⁡(2a)+1)sin⁡(−a+π4)\left(1 - \sin{\left(2 a \right)}\right) \cos{\left(- a + \frac{\pi}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \sin{\left(- a + \frac{\pi}{4} \right)} 
(1 - sin(2*a))*cos(pi/4 - a) - (1 + sin(2*a))*sin(pi/4 - a)
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