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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(a)       sin(a)  
---------- - ----------
1 - cos(a)   1 + cos(a)
−sin⁡(a)cos⁡(a)+1+sin⁡(a)1−cos⁡(a)- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} + \frac{\sin{\left(a \right)}}{1 - \cos{\left(a \right)}} 
sin(a)/(1 - cos(a)) - sin(a)/(1 + cos(a))
Simplificación general [src] 
  2   
------
tan(a)
2tan⁡(a)\frac{2}{\tan{\left(a \right)}} 
2/tan(a)
Denominador común [src] 
-2*cos(a)*sin(a)
----------------
          2     
  -1 + cos (a)
−2sin⁡(a)cos⁡(a)cos⁡2(a)−1- \frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\cos^{2}{\left(a \right)} - 1} 
-2*cos(a)*sin(a)/(-1 + cos(a)^2)
Unión de expresiones racionales [src] 
     2*cos(a)*sin(a)     
-------------------------
(1 - cos(a))*(1 + cos(a))
2sin⁡(a)cos⁡(a)(1−cos⁡(a))(cos⁡(a)+1)\frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\left(1 - \cos{\left(a \right)}\right) \left(\cos{\left(a \right)} + 1\right)} 
2*cos(a)*sin(a)/((1 - cos(a))*(1 + cos(a)))
Combinatoria [src] 
     -2*cos(a)*sin(a)     
--------------------------
(1 + cos(a))*(-1 + cos(a))
−2sin⁡(a)cos⁡(a)(cos⁡(a)−1)(cos⁡(a)+1)- \frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} + 1\right)} 
-2*cos(a)*sin(a)/((1 + cos(a))*(-1 + cos(a)))
Respuesta numérica [src] 
sin(a)/(1.0 - cos(a)) - sin(a)/(1.0 + cos(a))
sin(a)/(1.0 - cos(a)) - sin(a)/(1.0 + cos(a))
Denominador racional [src] 
(1 + cos(a))*sin(a) - (1 - cos(a))*sin(a)
-----------------------------------------
        (1 - cos(a))*(1 + cos(a))
−(1−cos⁡(a))sin⁡(a)+(cos⁡(a)+1)sin⁡(a)(1−cos⁡(a))(cos⁡(a)+1)\frac{- \left(1 - \cos{\left(a \right)}\right) \sin{\left(a \right)} + \left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}{\left(1 - \cos{\left(a \right)}\right) \left(\cos{\left(a \right)} + 1\right)} 
((1 + cos(a))*sin(a) - (1 - cos(a))*sin(a))/((1 - cos(a))*(1 + cos(a)))
Parte trigonométrica [src] 
2*cot(a)
2cot⁡(a)2 \cot{\left(a \right)} 
                   /a\                                /a\            
              2*cot|-|                           2*cot|-|            
                   \2/                                \2/            
- -------------------------------- + --------------------------------
                /            2/a\\                 /            2/a\\
                |    -1 + cot |-||                 |    -1 + cot |-||
  /       2/a\\ |             \2/|   /       2/a\\ |             \2/|
  |1 + cot |-||*|1 + ------------|   |1 + cot |-||*|1 - ------------|
  \        \2// |           2/a\ |   \        \2// |           2/a\ |
                |    1 + cot |-| |                 |    1 + cot |-| |
                \            \2/ /                 \            \2/ /
−2cot⁡(a2)(cot⁡2(a2)−1cot⁡2(a2)+1+1)(cot⁡2(a2)+1)+2cot⁡(a2)(−cot⁡2(a2)−1cot⁡2(a2)+1+1)(cot⁡2(a2)+1)- \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
                   /a\                               /a\           
              2*tan|-|                          2*tan|-|           
                   \2/                               \2/           
- ------------------------------- + -------------------------------
                /           2/a\\                 /           2/a\\
                |    1 - tan |-||                 |    1 - tan |-||
  /       2/a\\ |            \2/|   /       2/a\\ |            \2/|
  |1 + tan |-||*|1 + -----------|   |1 + tan |-||*|1 - -----------|
  \        \2// |           2/a\|   \        \2// |           2/a\|
                |    1 + tan |-||                 |    1 + tan |-||
                \            \2//                 \            \2//
−2tan⁡(a2)(1−tan⁡2(a2)tan⁡2(a2)+1+1)(tan⁡2(a2)+1)+2tan⁡(a2)(−1−tan⁡2(a2)tan⁡2(a2)+1+1)(tan⁡2(a2)+1)- \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
     sin(a)            sin(a)    
--------------- - ---------------
       /    pi\          /    pi\
1 - sin|a + --|   1 + sin|a + --|
       \    2 /          \    2 /
−sin⁡(a)sin⁡(a+π2)+1+sin⁡(a)1−sin⁡(a+π2)- \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} + 1} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a + \frac{\pi}{2} \right)}} 
         1                     1         
------------------- - -------------------
/      1   \          /      1   \       
|1 - ------|*csc(a)   |1 + ------|*csc(a)
\    sec(a)/          \    sec(a)/
−1(1+1sec⁡(a))csc⁡(a)+1(1−1sec⁡(a))csc⁡(a)- \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}} 
  2*csc(a) 
-----------
   /pi    \
csc|-- - a|
   \2     /
2csc⁡(a)csc⁡(−a+π2)\frac{2 \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} 
           1                          1            
------------------------ - ------------------------
/         1     \          /         1     \       
|1 - -----------|*csc(a)   |1 + -----------|*csc(a)
|       /pi    \|          |       /pi    \|       
|    csc|-- - a||          |    csc|-- - a||       
\       \2     //          \       \2     //
−1(1+1csc⁡(−a+π2))csc⁡(a)+1(1−1csc⁡(−a+π2))csc⁡(a)- \frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}} 
sin(2*a)
--------
   2    
sin (a)
sin⁡(2a)sin⁡2(a)\frac{\sin{\left(2 a \right)}}{\sin^{2}{\left(a \right)}} 
     /    pi\
2*sec|a - --|
     \    2 /
-------------
    sec(a)
2sec⁡(a−π2)sec⁡(a)\frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} 
   /    pi\      /    pi\
cos|a - --|   cos|a - --|
   \    2 /      \    2 /
----------- - -----------
 1 - cos(a)    1 + cos(a)
−cos⁡(a−π2)cos⁡(a)+1+cos⁡(a−π2)1−cos⁡(a)- \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} + 1} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{1 - \cos{\left(a \right)}} 
           1                          1            
------------------------ - ------------------------
/      1   \    /    pi\   /      1   \    /    pi\
|1 - ------|*sec|a - --|   |1 + ------|*sec|a - --|
\    sec(a)/    \    2 /   \    sec(a)/    \    2 /
−1(1+1sec⁡(a))sec⁡(a−π2)+1(1−1sec⁡(a))sec⁡(a−π2)- \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} 
  2   
------
tan(a)
2tan⁡(a)\frac{2}{\tan{\left(a \right)}} 
  2*cos(a) 
-----------
   /    pi\
cos|a - --|
   \    2 /
2cos⁡(a)cos⁡(a−π2)\frac{2 \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} 
2*cos(a)/cos(a - pi/2)
Potencias [src] 
   /   -I*a    I*a\       /   -I*a    I*a\ 
 I*\- e     + e   /     I*\- e     + e   / 
-------------------- - --------------------
  /     I*a    -I*a\     /     I*a    -I*a\
  |    e      e    |     |    e      e    |
2*|1 + ---- + -----|   2*|1 - ---- - -----|
  \     2       2  /     \     2       2  /
i(eia−e−ia)2(eia2+1+e−ia2)−i(eia−e−ia)2(−eia2+1−e−ia2)\frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{e^{i a}}{2} + 1 + \frac{e^{- i a}}{2}\right)} - \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{e^{i a}}{2} + 1 - \frac{e^{- i a}}{2}\right)} 
i*(-exp(-i*a) + exp(i*a))/(2*(1 + exp(i*a)/2 + exp(-i*a)/2)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 - exp(i*a)/2 - exp(-i*a)/2))
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