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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(a)*cos(a)  
-----------------
   / 2\      / 2\
sin\a / - cos\a /
sin⁡(a)cos⁡(a)sin⁡(a2)−cos⁡(a2)\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a^{2} \right)} - \cos{\left(a^{2} \right)}} 
(sin(a)*cos(a))/(sin(a^2) - cos(a^2))
Simplificación general [src] 
   ___          
-\/ 2 *sin(2*a) 
----------------
      / 2   pi\ 
 4*cos|a  + --| 
      \     4 /
−2sin⁡(2a)4cos⁡(a2+π4)- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}} 
-sqrt(2)*sin(2*a)/(4*cos(a^2 + pi/4))
Combinatoria [src] 
  -cos(a)*sin(a)   
-------------------
     / 2\      / 2\
- sin\a / + cos\a /
−sin⁡(a)cos⁡(a)−sin⁡(a2)+cos⁡(a2)- \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{- \sin{\left(a^{2} \right)} + \cos{\left(a^{2} \right)}} 
-cos(a)*sin(a)/(-sin(a^2) + cos(a^2))
Denominador común [src] 
  -cos(a)*sin(a)   
-------------------
     / 2\      / 2\
- sin\a / + cos\a /
−sin⁡(a)cos⁡(a)−sin⁡(a2)+cos⁡(a2)- \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{- \sin{\left(a^{2} \right)} + \cos{\left(a^{2} \right)}} 
-cos(a)*sin(a)/(-sin(a^2) + cos(a^2))
Potencias [src] 
       / I*a    -I*a\                      
       |e      e    | /   -I*a    I*a\     
    -I*|---- + -----|*\- e     + e   /     
       \ 2       2  /                      
-------------------------------------------
  /      2        2     /       2       2\\
  |   I*a     -I*a      |   -I*a     I*a ||
  |  e       e        I*\- e      + e    /|
2*|- ----- - ------ - --------------------|
  \    2       2               2          /
−i(eia2+e−ia2)(eia−e−ia)2(−i(eia2−e−ia2)2−eia22−e−ia22)- \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{i \left(e^{i a^{2}} - e^{- i a^{2}}\right)}{2} - \frac{e^{i a^{2}}}{2} - \frac{e^{- i a^{2}}}{2}\right)} 
-i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(-i*a) + exp(i*a))/(2*(-exp(i*a^2)/2 - exp(-i*a^2)/2 - i*(-exp(-i*a^2) + exp(i*a^2))/2))
Parte trigonométrica [src] 
   ___    /   2   pi\ 
-\/ 2 *csc|- a  + --| 
          \       4 / 
----------------------
      4*csc(2*a)
−2csc⁡(−a2+π4)4csc⁡(2a)- \frac{\sqrt{2} \csc{\left(- a^{2} + \frac{\pi}{4} \right)}}{4 \csc{\left(2 a \right)}} 
   ___    / 2   pi\ 
-\/ 2 *sec|a  + --| 
          \     4 / 
--------------------
       /      pi\   
  4*sec|2*a - --|   
       \      2 /
−2sec⁡(a2+π4)4sec⁡(2a−π2)- \frac{\sqrt{2} \sec{\left(a^{2} + \frac{\pi}{4} \right)}}{4 \sec{\left(2 a - \frac{\pi}{2} \right)}} 
               /        / 2     \\        
   ___    2    |       2|a    pi||        
-\/ 2 *cos (a)*|1 + tan |-- + --||*tan(a) 
               \        \2    8 //        
------------------------------------------
            /        / 2     \\           
            |       2|a    pi||           
          2*|1 - tan |-- + --||           
            \        \2    8 //
−2(tan⁡2(a22+π8)+1)cos⁡2(a)tan⁡(a)2(1−tan⁡2(a22+π8))- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(a \right)} \tan{\left(a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)}\right)} 
   ___          
-\/ 2 *sin(2*a) 
----------------
     / 2   3*pi\
4*sin|a  + ----|
     \      4  /
−2sin⁡(2a)4sin⁡(a2+3π4)- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \sin{\left(a^{2} + \frac{3 \pi}{4} \right)}} 
              /        2/a\\    /a\            
            2*|-1 + cot |-||*cot|-|            
              \         \2//    \2/            
-----------------------------------------------
               /           / 2\         / 2\  \
               |          2|a |         |a |  |
             2 |  -1 + cot |--|    2*cot|--|  |
/       2/a\\  |           \2 /         \2 /  |
|1 + cot |-|| *|- ------------- + ------------|
\        \2//  |           / 2\           / 2\|
               |          2|a |          2|a ||
               |   1 + cot |--|   1 + cot |--||
               \           \2 /           \2 //
2(cot⁡2(a2)−1)cot⁡(a2)(−cot⁡2(a22)−1cot⁡2(a22)+1+2cot⁡(a22)cot⁡2(a22)+1)(cot⁡2(a2)+1)2\frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{a^{2}}{2} \right)} - 1}{\cot^{2}{\left(\frac{a^{2}}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a^{2}}{2} \right)}}{\cot^{2}{\left(\frac{a^{2}}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} 
   ___          
-\/ 2 *sin(2*a) 
----------------
      / 2   pi\ 
 4*cos|a  + --| 
      \     4 /
−2sin⁡(2a)4cos⁡(a2+π4)- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}} 
                         ___                               
                       \/ 2 *sin(2*a)                      
-----------------------------------------------------------
  /                     ___________    ___________        \
  |    ___    / 2\     /       ___    /       ___     / 2\|
2*\- \/ 2 *cos\a / + \/  2 + \/ 2  *\/  2 - \/ 2  *sin\a //
2sin⁡(2a)2(2−22+2sin⁡(a2)−2cos⁡(a2))\frac{\sqrt{2} \sin{\left(2 a \right)}}{2 \left(\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(a^{2} \right)} - \sqrt{2} \cos{\left(a^{2} \right)}\right)} 
                     1                     
-------------------------------------------
/   1           1      \           /pi    \
|------- - ------------|*csc(a)*csc|-- - a|
|   / 2\      /pi    2\|           \2     /
|csc\a /   csc|-- - a ||                   
\             \2      //
1(−1csc⁡(−a2+π2)+1csc⁡(a2))csc⁡(a)csc⁡(−a+π2)\frac{1}{\left(- \frac{1}{\csc{\left(- a^{2} + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(a^{2} \right)}}\right) \csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}} 
        /        / 2     \\        
    ___ |       2|a    pi||        
 -\/ 2 *|1 + tan |-- + --||*tan(a) 
        \        \2    8 //        
-----------------------------------
                /        / 2     \\
  /       2   \ |       2|a    pi||
2*\1 + tan (a)/*|1 - tan |-- + --||
                \        \2    8 //
−2(tan⁡2(a22+π8)+1)tan⁡(a)2(1−tan⁡2(a22+π8))(tan⁡2(a)+1)- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \tan{\left(a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)} 
                1                
---------------------------------
/   1         1   \              
|------- - -------|*csc(a)*sec(a)
|   / 2\      / 2\|              
\csc\a /   sec\a //
1(−1sec⁡(a2)+1csc⁡(a2))csc⁡(a)sec⁡(a)\frac{1}{\left(- \frac{1}{\sec{\left(a^{2} \right)}} + \frac{1}{\csc{\left(a^{2} \right)}}\right) \csc{\left(a \right)} \sec{\left(a \right)}} 
   ___    /   2   pi\          
-\/ 2 *csc|- a  + --|*sin(2*a) 
          \       4 /          
-------------------------------
               4
−2sin⁡(2a)csc⁡(−a2+π4)4- \frac{\sqrt{2} \sin{\left(2 a \right)} \csc{\left(- a^{2} + \frac{\pi}{4} \right)}}{4} 
                     1                     
-------------------------------------------
/     1            1   \           /    pi\
|------------ - -------|*sec(a)*sec|a - --|
|   / 2   pi\      / 2\|           \    2 /
|sec|a  - --|   sec\a /|                   
\   \     2 /          /
1(1sec⁡(a2−π2)−1sec⁡(a2))sec⁡(a)sec⁡(a−π2)\frac{1}{\left(\frac{1}{\sec{\left(a^{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(a^{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(a - \frac{\pi}{2} \right)}} 
   ___    / 2   pi\          
-\/ 2 *sec|a  + --|*sin(2*a) 
          \     4 /          
-----------------------------
              4
−2sin⁡(2a)sec⁡(a2+π4)4- \frac{\sqrt{2} \sin{\left(2 a \right)} \sec{\left(a^{2} + \frac{\pi}{4} \right)}}{4} 
        /        / 2     \\         
    ___ |       2|a    pi||         
 -\/ 2 *|1 + cot |-- + --||*cot(a)  
        \        \2    8 //         
------------------------------------
                /         / 2     \\
  /       2   \ |        2|a    pi||
2*\1 + cot (a)/*|-1 + cot |-- + --||
                \         \2    8 //
−2(cot⁡2(a22+π8)+1)cot⁡(a)2(cot⁡2(a)+1)(cot⁡2(a22+π8)−1)- \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \cot{\left(a \right)}}{2 \left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} - 1\right)} 
              /       2/a\\    /a\            
            2*|1 - tan |-||*tan|-|            
              \        \2//    \2/            
----------------------------------------------
               /          / 2\         / 2\  \
               |         2|a |         |a |  |
             2 |  1 - tan |--|    2*tan|--|  |
/       2/a\\  |          \2 /         \2 /  |
|1 + tan |-|| *|- ------------ + ------------|
\        \2//  |          / 2\           / 2\|
               |         2|a |          2|a ||
               |  1 + tan |--|   1 + tan |--||
               \          \2 /           \2 //
2(1−tan⁡2(a2))tan⁡(a2)(−1−tan⁡2(a22)tan⁡2(a22)+1+2tan⁡(a22)tan⁡2(a22)+1)(tan⁡2(a2)+1)2\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{a^{2}}{2} \right)}}{\tan^{2}{\left(\frac{a^{2}}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a^{2}}{2} \right)}}{\tan^{2}{\left(\frac{a^{2}}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} 
             /    pi\   
   sin(a)*sin|a + --|   
             \    2 /   
------------------------
     / 2   pi\      / 2\
- sin|a  + --| + sin\a /
     \     2 /
sin⁡(a)sin⁡(a+π2)sin⁡(a2)−sin⁡(a2+π2)\frac{\sin{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a^{2} \right)} - \sin{\left(a^{2} + \frac{\pi}{2} \right)}} 
             /    pi\   
   cos(a)*cos|a - --|   
             \    2 /   
------------------------
     / 2\      / 2   pi\
- cos\a / + cos|a  - --|
               \     2 /
cos⁡(a)cos⁡(a−π2)−cos⁡(a2)+cos⁡(a2−π2)\frac{\cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{- \cos{\left(a^{2} \right)} + \cos{\left(a^{2} - \frac{\pi}{2} \right)}} 
   ___    /      pi\ 
-\/ 2 *cos|2*a - --| 
          \      2 / 
---------------------
         / 2   pi\   
    4*cos|a  + --|   
         \     4 /
−2cos⁡(2a−π2)4cos⁡(a2+π4)- \frac{\sqrt{2} \cos{\left(2 a - \frac{\pi}{2} \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}} 
-sqrt(2)*cos(2*a - pi/2)/(4*cos(a^2 + pi/4))
Respuesta numérica [src] 
cos(a)*sin(a)/(-cos(a^2) + sin(a^2))
cos(a)*sin(a)/(-cos(a^2) + sin(a^2))
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