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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(2*a)/(cos(2*a)+sin(2*a))

¿Cómo vas a descomponer esta sin(2*a)/(cos(2*a)+sin(2*a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
      sin(2*a)     
-------------------
cos(2*a) + sin(2*a)
sin⁡(2a)sin⁡(2a)+cos⁡(2a)\frac{\sin{\left(2 a \right)}}{\sin{\left(2 a \right)} + \cos{\left(2 a \right)}} 
sin(2*a)/(cos(2*a) + sin(2*a))
Simplificación general [src] 
   ___         
 \/ 2 *sin(2*a)
---------------
     /      pi\
2*sin|2*a + --|
     \      4 /
2sin⁡(2a)2sin⁡(2a+π4)\frac{\sqrt{2} \sin{\left(2 a \right)}}{2 \sin{\left(2 a + \frac{\pi}{4} \right)}} 
sqrt(2)*sin(2*a)/(2*sin(2*a + pi/4))
Potencias [src] 
              /   -2*I*a    2*I*a\           
           -I*\- e       + e     /           
---------------------------------------------
  / -2*I*a    2*I*a     /   -2*I*a    2*I*a\\
  |e         e        I*\- e       + e     /|
2*|------- + ------ - ----------------------|
  \   2        2                2           /
−i(e2ia−e−2ia)2(−i(e2ia−e−2ia)2+e2ia2+e−2ia2)- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2 \left(- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right)} 
-i*(-exp(-2*i*a) + exp(2*i*a))/(2*(exp(-2*i*a)/2 + exp(2*i*a)/2 - i*(-exp(-2*i*a) + exp(2*i*a))/2))
Abrimos la expresión [src] 
        2*cos(a)*sin(a)         
--------------------------------
          2                     
-1 + 2*cos (a) + 2*cos(a)*sin(a)
2sin⁡(a)cos⁡(a)2sin⁡(a)cos⁡(a)+2cos⁡2(a)−1\frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1} 
2*cos(a)*sin(a)/(-1 + 2*cos(a)^2 + 2*cos(a)*sin(a))
Parte trigonométrica [src] 
   ___         
 \/ 2 *sin(2*a)
---------------
     /      pi\
2*sin|2*a + --|
     \      4 /
2sin⁡(2a)2sin⁡(2a+π4)\frac{\sqrt{2} \sin{\left(2 a \right)}}{2 \sin{\left(2 a + \frac{\pi}{4} \right)}} 
        /      pi\      
     cos|2*a - --|      
        \      2 /      
------------------------
              /      pi\
cos(2*a) + cos|2*a - --|
              \      2 /
cos⁡(2a−π2)cos⁡(2a)+cos⁡(2a−π2)\frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{\cos{\left(2 a \right)} + \cos{\left(2 a - \frac{\pi}{2} \right)}} 
                   1                    
----------------------------------------
/   1             1      \    /      pi\
|-------- + -------------|*sec|2*a - --|
|sec(2*a)      /      pi\|    \      2 /
|           sec|2*a - --||              
\              \      2 //
1(1sec⁡(2a−π2)+1sec⁡(2a))sec⁡(2a−π2)\frac{1}{\left(\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}\right) \sec{\left(2 a - \frac{\pi}{2} \right)}} 
        sin(2*a)        
------------------------
              /pi      \
sin(2*a) + sin|-- + 2*a|
              \2       /
sin⁡(2a)sin⁡(2a)+sin⁡(2a+π2)\frac{\sin{\left(2 a \right)}}{\sin{\left(2 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}} 
                 2*tan(a)                
-----------------------------------------
              /       2                 \
/       2   \ |1 - tan (a)     2*tan(a) |
\1 + tan (a)/*|----------- + -----------|
              |       2             2   |
              \1 + tan (a)   1 + tan (a)/
2tan⁡(a)(1−tan⁡2(a)tan⁡2(a)+1+2tan⁡(a)tan⁡2(a)+1)(tan⁡2(a)+1)\frac{2 \tan{\left(a \right)}}{\left(\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}\right) \left(\tan^{2}{\left(a \right)} + 1\right)} 
  ___    2              /    pi\
\/ 2 *sin (a)*cot(a)*tan|a + --|
                        \    8 /
--------------------------------
              2/    pi\         
         2*sin |a + --|         
               \    8 /
2sin⁡2(a)tan⁡(a+π8)cot⁡(a)2sin⁡2(a+π8)\frac{\sqrt{2} \sin^{2}{\left(a \right)} \tan{\left(a + \frac{\pi}{8} \right)} \cot{\left(a \right)}}{2 \sin^{2}{\left(a + \frac{\pi}{8} \right)}} 
  ___    /      pi\         
\/ 2 *csc|2*a + --|*sin(2*a)
         \      4 /         
----------------------------
             2
2sin⁡(2a)csc⁡(2a+π4)2\frac{\sqrt{2} \sin{\left(2 a \right)} \csc{\left(2 a + \frac{\pi}{4} \right)}}{2} 
                 1                 
-----------------------------------
/   1             1      \         
|-------- + -------------|*csc(2*a)
|csc(2*a)      /pi      \|         
|           csc|-- - 2*a||         
\              \2       //
1(1csc⁡(−2a+π2)+1csc⁡(2a))csc⁡(2a)\frac{1}{\left(\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 a \right)}}\right) \csc{\left(2 a \right)}} 
              1               
------------------------------
/   1          1    \         
|-------- + --------|*csc(2*a)
\csc(2*a)   sec(2*a)/
1(1sec⁡(2a)+1csc⁡(2a))csc⁡(2a)\frac{1}{\left(\frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(2 a \right)}}\right) \csc{\left(2 a \right)}} 
  ___    /      pi\         
\/ 2 *sec|2*a - --|*sin(2*a)
         \      4 /         
----------------------------
             2
2sin⁡(2a)sec⁡(2a−π4)2\frac{\sqrt{2} \sin{\left(2 a \right)} \sec{\left(2 a - \frac{\pi}{4} \right)}}{2} 
  ___ /       2/    pi\\       
\/ 2 *|1 + cot |a + --||*cot(a)
      \        \    8 //       
-------------------------------
    /       2   \    /    pi\  
  2*\1 + cot (a)/*cot|a + --|  
                     \    8 /
2(cot⁡2(a+π8)+1)cot⁡(a)2(cot⁡2(a)+1)cot⁡(a+π8)\frac{\sqrt{2} \left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \cot{\left(a \right)}}{2 \left(\cot^{2}{\left(a \right)} + 1\right) \cot{\left(a + \frac{\pi}{8} \right)}} 
   ___         
 \/ 2 *sin(2*a)
---------------
     /      pi\
2*cos|2*a - --|
     \      4 /
2sin⁡(2a)2cos⁡(2a−π4)\frac{\sqrt{2} \sin{\left(2 a \right)}}{2 \cos{\left(2 a - \frac{\pi}{4} \right)}} 
  ___    2    /       2/    pi\\       
\/ 2 *cos (a)*|1 + tan |a + --||*tan(a)
              \        \    8 //       
---------------------------------------
                  /    pi\             
             2*tan|a + --|             
                  \    8 /
2(tan⁡2(a+π8)+1)cos⁡2(a)tan⁡(a)2tan⁡(a+π8)\frac{\sqrt{2} \left(\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(a \right)} \tan{\left(a \right)}}{2 \tan{\left(a + \frac{\pi}{8} \right)}} 
                 2*cot(a)                 
------------------------------------------
              /        2                 \
/       2   \ |-1 + cot (a)     2*cot(a) |
\1 + cot (a)/*|------------ + -----------|
              |       2              2   |
              \1 + cot (a)    1 + cot (a)/
2cot⁡(a)(cot⁡2(a)−1cot⁡2(a)+1+2cot⁡(a)cot⁡2(a)+1)(cot⁡2(a)+1)\frac{2 \cot{\left(a \right)}}{\left(\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + \frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}\right) \left(\cot^{2}{\left(a \right)} + 1\right)} 
  ___ /       2/    pi\\       
\/ 2 *|1 + tan |a + --||*tan(a)
      \        \    8 //       
-------------------------------
    /       2   \    /    pi\  
  2*\1 + tan (a)/*tan|a + --|  
                     \    8 /
2(tan⁡2(a+π8)+1)tan⁡(a)2(tan⁡2(a)+1)tan⁡(a+π8)\frac{\sqrt{2} \left(\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1\right) \tan{\left(a \right)}}{2 \left(\tan^{2}{\left(a \right)} + 1\right) \tan{\left(a + \frac{\pi}{8} \right)}} 
  ___    /      pi\
\/ 2 *csc|2*a + --|
         \      4 /
-------------------
     2*csc(2*a)
2csc⁡(2a+π4)2csc⁡(2a)\frac{\sqrt{2} \csc{\left(2 a + \frac{\pi}{4} \right)}}{2 \csc{\left(2 a \right)}} 
  ___    /      pi\
\/ 2 *cos|2*a - --|
         \      2 /
-------------------
       /      pi\  
  2*cos|2*a - --|  
       \      4 /
2cos⁡(2a−π2)2cos⁡(2a−π4)\frac{\sqrt{2} \cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a - \frac{\pi}{4} \right)}} 
  ___    /      pi\
\/ 2 *sec|2*a - --|
         \      4 /
-------------------
       /      pi\  
  2*sec|2*a - --|  
       \      2 /
2sec⁡(2a−π4)2sec⁡(2a−π2)\frac{\sqrt{2} \sec{\left(2 a - \frac{\pi}{4} \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} 
sqrt(2)*sec(2*a - pi/4)/(2*sec(2*a - pi/2))
Respuesta numérica [src] 
sin(2*a)/(cos(2*a) + sin(2*a))
sin(2*a)/(cos(2*a) + sin(2*a))
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