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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(l)/(1+cos(l))+sin(l)/((1)-cos(l))

¿Cómo vas a descomponer esta sin(l)/(1+cos(l))+sin(l)/((1)-cos(l)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(l)       sin(l)  
---------- + ----------
1 + cos(l)   1 - cos(l)
sin⁡(l)cos⁡(l)+1+sin⁡(l)1−cos⁡(l)\frac{\sin{\left(l \right)}}{\cos{\left(l \right)} + 1} + \frac{\sin{\left(l \right)}}{1 - \cos{\left(l \right)}} 
sin(l)/(1 + cos(l)) + sin(l)/(1 - cos(l))
Simplificación general [src] 
  2   
------
sin(l)
2sin⁡(l)\frac{2}{\sin{\left(l \right)}} 
2/sin(l)
Respuesta numérica [src] 
sin(l)/(1.0 - cos(l)) + sin(l)/(1.0 + cos(l))
sin(l)/(1.0 - cos(l)) + sin(l)/(1.0 + cos(l))
Denominador racional [src] 
(1 - cos(l))*sin(l) + (1 + cos(l))*sin(l)
-----------------------------------------
        (1 - cos(l))*(1 + cos(l))
(1−cos⁡(l))sin⁡(l)+(cos⁡(l)+1)sin⁡(l)(1−cos⁡(l))(cos⁡(l)+1)\frac{\left(1 - \cos{\left(l \right)}\right) \sin{\left(l \right)} + \left(\cos{\left(l \right)} + 1\right) \sin{\left(l \right)}}{\left(1 - \cos{\left(l \right)}\right) \left(\cos{\left(l \right)} + 1\right)} 
((1 - cos(l))*sin(l) + (1 + cos(l))*sin(l))/((1 - cos(l))*(1 + cos(l)))
Unión de expresiones racionales [src] 
         2*sin(l)        
-------------------------
(1 - cos(l))*(1 + cos(l))
2sin⁡(l)(1−cos⁡(l))(cos⁡(l)+1)\frac{2 \sin{\left(l \right)}}{\left(1 - \cos{\left(l \right)}\right) \left(\cos{\left(l \right)} + 1\right)} 
2*sin(l)/((1 - cos(l))*(1 + cos(l)))
Combinatoria [src] 
        -2*sin(l)         
--------------------------
(1 + cos(l))*(-1 + cos(l))
−2sin⁡(l)(cos⁡(l)−1)(cos⁡(l)+1)- \frac{2 \sin{\left(l \right)}}{\left(\cos{\left(l \right)} - 1\right) \left(\cos{\left(l \right)} + 1\right)} 
-2*sin(l)/((1 + cos(l))*(-1 + cos(l)))
Potencias [src] 
     /   -I*l    I*l\       /   -I*l    I*l\ 
   I*\- e     + e   /     I*\- e     + e   / 
- -------------------- - --------------------
    /     I*l    -I*l\     /     I*l    -I*l\
    |    e      e    |     |    e      e    |
  2*|1 + ---- + -----|   2*|1 - ---- - -----|
    \     2       2  /     \     2       2  /
−i(eil−e−il)2(eil2+1+e−il2)−i(eil−e−il)2(−eil2+1−e−il2)- \frac{i \left(e^{i l} - e^{- i l}\right)}{2 \left(\frac{e^{i l}}{2} + 1 + \frac{e^{- i l}}{2}\right)} - \frac{i \left(e^{i l} - e^{- i l}\right)}{2 \left(- \frac{e^{i l}}{2} + 1 - \frac{e^{- i l}}{2}\right)} 
-i*(-exp(-i*l) + exp(i*l))/(2*(1 + exp(i*l)/2 + exp(-i*l)/2)) - i*(-exp(-i*l) + exp(i*l))/(2*(1 - exp(i*l)/2 - exp(-i*l)/2))
Denominador común [src] 
 -2*sin(l)  
------------
        2   
-1 + cos (l)
−2sin⁡(l)cos⁡2(l)−1- \frac{2 \sin{\left(l \right)}}{\cos^{2}{\left(l \right)} - 1} 
-2*sin(l)/(-1 + cos(l)^2)
Parte trigonométrica [src] 
                 /l\                               /l\           
            2*tan|-|                          2*tan|-|           
                 \2/                               \2/           
------------------------------- + -------------------------------
              /           2/l\\                 /           2/l\\
              |    1 - tan |-||                 |    1 - tan |-||
/       2/l\\ |            \2/|   /       2/l\\ |            \2/|
|1 + tan |-||*|1 + -----------|   |1 + tan |-||*|1 - -----------|
\        \2// |           2/l\|   \        \2// |           2/l\|
              |    1 + tan |-||                 |    1 + tan |-||
              \            \2//                 \            \2//
2tan⁡(l2)(1−tan⁡2(l2)tan⁡2(l2)+1+1)(tan⁡2(l2)+1)+2tan⁡(l2)(−1−tan⁡2(l2)tan⁡2(l2)+1+1)(tan⁡2(l2)+1)\frac{2 \tan{\left(\frac{l}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{l}{2} \right)}}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{l}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{l}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{l}{2} \right)}}{\tan^{2}{\left(\frac{l}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{l}{2} \right)} + 1\right)} 
     2     
-----------
   /    pi\
cos|l - --|
   \    2 /
2cos⁡(l−π2)\frac{2}{\cos{\left(l - \frac{\pi}{2} \right)}} 
       2/l\
1 + cot |-|
        \2/
-----------
      /l\  
   cot|-|  
      \2/
cot⁡2(l2)+1cot⁡(l2)\frac{\cot^{2}{\left(\frac{l}{2} \right)} + 1}{\cot{\left(\frac{l}{2} \right)}} 
     sin(l)            sin(l)    
--------------- + ---------------
       /    pi\          /    pi\
1 - sin|l + --|   1 + sin|l + --|
       \    2 /          \    2 /
sin⁡(l)sin⁡(l+π2)+1+sin⁡(l)1−sin⁡(l+π2)\frac{\sin{\left(l \right)}}{\sin{\left(l + \frac{\pi}{2} \right)} + 1} + \frac{\sin{\left(l \right)}}{1 - \sin{\left(l + \frac{\pi}{2} \right)}} 
   /    pi\      /    pi\
cos|l - --|   cos|l - --|
   \    2 /      \    2 /
----------- + -----------
 1 - cos(l)    1 + cos(l)
cos⁡(l−π2)cos⁡(l)+1+cos⁡(l−π2)1−cos⁡(l)\frac{\cos{\left(l - \frac{\pi}{2} \right)}}{\cos{\left(l \right)} + 1} + \frac{\cos{\left(l - \frac{\pi}{2} \right)}}{1 - \cos{\left(l \right)}} 
         1                     1         
------------------- + -------------------
/      1   \          /      1   \       
|1 + ------|*csc(l)   |1 - ------|*csc(l)
\    sec(l)/          \    sec(l)/
1(1+1sec⁡(l))csc⁡(l)+1(1−1sec⁡(l))csc⁡(l)\frac{1}{\left(1 + \frac{1}{\sec{\left(l \right)}}\right) \csc{\left(l \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(l \right)}}\right) \csc{\left(l \right)}} 
  2   
------
sin(l)
2sin⁡(l)\frac{2}{\sin{\left(l \right)}} 
2*csc(l)
2csc⁡(l)2 \csc{\left(l \right)} 
           1                          1            
------------------------ + ------------------------
/      1   \    /    pi\   /      1   \    /    pi\
|1 + ------|*sec|l - --|   |1 - ------|*sec|l - --|
\    sec(l)/    \    2 /   \    sec(l)/    \    2 /
1(1+1sec⁡(l))sec⁡(l−π2)+1(1−1sec⁡(l))sec⁡(l−π2)\frac{1}{\left(1 + \frac{1}{\sec{\left(l \right)}}\right) \sec{\left(l - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(l \right)}}\right) \sec{\left(l - \frac{\pi}{2} \right)}} 
       2/l\
1 + tan |-|
        \2/
-----------
      /l\  
   tan|-|  
      \2/
tan⁡2(l2)+1tan⁡(l2)\frac{\tan^{2}{\left(\frac{l}{2} \right)} + 1}{\tan{\left(\frac{l}{2} \right)}} 
           1                          1            
------------------------ + ------------------------
/         1     \          /         1     \       
|1 + -----------|*csc(l)   |1 - -----------|*csc(l)
|       /pi    \|          |       /pi    \|       
|    csc|-- - l||          |    csc|-- - l||       
\       \2     //          \       \2     //
1(1+1csc⁡(−l+π2))csc⁡(l)+1(1−1csc⁡(−l+π2))csc⁡(l)\frac{1}{\left(1 + \frac{1}{\csc{\left(- l + \frac{\pi}{2} \right)}}\right) \csc{\left(l \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- l + \frac{\pi}{2} \right)}}\right) \csc{\left(l \right)}} 
     /    pi\
2*sec|l - --|
     \    2 /
2sec⁡(l−π2)2 \sec{\left(l - \frac{\pi}{2} \right)} 
                 /l\                                /l\            
            2*cot|-|                           2*cot|-|            
                 \2/                                \2/            
-------------------------------- + --------------------------------
              /            2/l\\                 /            2/l\\
              |    -1 + cot |-||                 |    -1 + cot |-||
/       2/l\\ |             \2/|   /       2/l\\ |             \2/|
|1 + cot |-||*|1 + ------------|   |1 + cot |-||*|1 - ------------|
\        \2// |           2/l\ |   \        \2// |           2/l\ |
              |    1 + cot |-| |                 |    1 + cot |-| |
              \            \2/ /                 \            \2/ /
2cot⁡(l2)(cot⁡2(l2)−1cot⁡2(l2)+1+1)(cot⁡2(l2)+1)+2cot⁡(l2)(−cot⁡2(l2)−1cot⁡2(l2)+1+1)(cot⁡2(l2)+1)\frac{2 \cot{\left(\frac{l}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{l}{2} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{l}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{l}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{l}{2} \right)} - 1}{\cot^{2}{\left(\frac{l}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{l}{2} \right)} + 1\right)} 
2*cot(l/2)/((1 + cot(l/2)^2)*(1 + (-1 + cot(l/2)^2)/(1 + cot(l/2)^2))) + 2*cot(l/2)/((1 + cot(l/2)^2)*(1 - (-1 + cot(l/2)^2)/(1 + cot(l/2)^2)))
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