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

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

Expresión a simplificar:

Solución

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