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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
       /  sin(x)       1   \
sin(x)*|---------- - ------|
       \1 - cos(x)   tan(x)/
(−1tan⁡(x)+sin⁡(x)1−cos⁡(x))sin⁡(x)\left(- \frac{1}{\tan{\left(x \right)}} + \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) \sin{\left(x \right)} 
sin(x)*(sin(x)/(1 - cos(x)) - 1/tan(x))
Simplificación general [src] 
1
11 
1
Respuesta numérica [src] 
(-1/tan(x) + sin(x)/(1.0 - cos(x)))*sin(x)
(-1/tan(x) + sin(x)/(1.0 - cos(x)))*sin(x)
Combinatoria [src] 
-(-1 + sin(x)*tan(x) + cos(x))*sin(x) 
--------------------------------------
         (-1 + cos(x))*tan(x)
−(sin⁡(x)tan⁡(x)+cos⁡(x)−1)sin⁡(x)(cos⁡(x)−1)tan⁡(x)- \frac{\left(\sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \tan{\left(x \right)}} 
-(-1 + sin(x)*tan(x) + cos(x))*sin(x)/((-1 + cos(x))*tan(x))
Potencias [src] 
                    /  / I*x    -I*x\      /   -I*x    I*x\ \ 
   /   -I*x    I*x\ |I*\e    + e    /    I*\- e     + e   / | 
-I*\- e     + e   /*|---------------- - --------------------| 
                    |    I*x    -I*x      /     I*x    -I*x\| 
                    | - e    + e          |    e      e    || 
                    |                   2*|1 - ---- - -----|| 
                    \                     \     2       2  // 
--------------------------------------------------------------
                              2
−i(−i(eix−e−ix)2(−eix2+1−e−ix2)+i(eix+e−ix)−eix+e−ix)(eix−e−ix)2- \frac{i \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)} + \frac{i \left(e^{i x} + e^{- i x}\right)}{- e^{i x} + e^{- i x}}\right) \left(e^{i x} - e^{- i x}\right)}{2} 
-i*(-exp(-i*x) + exp(i*x))*(i*(exp(i*x) + exp(-i*x))/(-exp(i*x) + exp(-i*x)) - i*(-exp(-i*x) + exp(i*x))/(2*(1 - exp(i*x)/2 - exp(-i*x)/2)))/2
Unión de expresiones racionales [src] 
(-1 + sin(x)*tan(x) + cos(x))*sin(x)
------------------------------------
        (1 - cos(x))*tan(x)
(sin⁡(x)tan⁡(x)+cos⁡(x)−1)sin⁡(x)(1−cos⁡(x))tan⁡(x)\frac{\left(\sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \tan{\left(x \right)}} 
(-1 + sin(x)*tan(x) + cos(x))*sin(x)/((1 - cos(x))*tan(x))
Denominador racional [src] 
(-1 + sin(x)*tan(x) + cos(x))*sin(x)
------------------------------------
        (1 - cos(x))*tan(x)
(sin⁡(x)tan⁡(x)+cos⁡(x)−1)sin⁡(x)(1−cos⁡(x))tan⁡(x)\frac{\left(\sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \tan{\left(x \right)}} 
(-1 + sin(x)*tan(x) + cos(x))*sin(x)/((1 - cos(x))*tan(x))
Abrimos la expresión [src] 
    2              
 sin (x)     sin(x)
---------- - ------
1 - cos(x)   tan(x)
−sin⁡(x)tan⁡(x)+sin⁡2(x)1−cos⁡(x)- \frac{\sin{\left(x \right)}}{\tan{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)}}{1 - \cos{\left(x \right)}} 
sin(x)^2/(1 - cos(x)) - sin(x)/tan(x)
Denominador común [src] 
 /             2                          \ 
-\-sin(x) + sin (x)*tan(x) + cos(x)*sin(x)/ 
--------------------------------------------
          -tan(x) + cos(x)*tan(x)
−sin⁡2(x)tan⁡(x)+sin⁡(x)cos⁡(x)−sin⁡(x)cos⁡(x)tan⁡(x)−tan⁡(x)- \frac{\sin^{2}{\left(x \right)} \tan{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)} - \tan{\left(x \right)}} 
-(-sin(x) + sin(x)^2*tan(x) + cos(x)*sin(x))/(-tan(x) + cos(x)*tan(x))
Parte trigonométrica [src] 
       1              sin(2*x)    
--------------- - ----------------
       /    pi\               2/x\
1 - sin|x + --|   4*sin(x)*sin |-|
       \    2 /                \2/
−sin⁡(2x)4sin⁡2(x2)sin⁡(x)+11−sin⁡(x+π2)- \frac{\sin{\left(2 x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}} + \frac{1}{1 - \sin{\left(x + \frac{\pi}{2} \right)}} 
       1                     /x\
---------------- - cot(x)*cot|-|
            2/x\             \2/
    -1 + cot |-|                
             \2/                
1 - ------------                
           2/x\                 
    1 + cot |-|                 
            \2/
−cot⁡(x2)cot⁡(x)+1−cot⁡2(x2)−1cot⁡2(x2)+1+1- \cot{\left(\frac{x}{2} \right)} \cot{\left(x \right)} + \frac{1}{- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1} 
1
11 
                              /    pi\
                           sec|x - --|
           1                  \    2 /
------------------------ - -----------
/      1   \    /    pi\      sec(x)  
|1 - ------|*sec|x - --|              
\    sec(x)/    \    2 /              
--------------------------------------
                /    pi\              
             sec|x - --|              
                \    2 /
−sec⁡(x−π2)sec⁡(x)+1(1−1sec⁡(x))sec⁡(x−π2)sec⁡(x−π2)\frac{- \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}}{\sec{\left(x - \frac{\pi}{2} \right)}} 
/   /    pi\              \            
|cos|x - --|              |            
|   \    2 /      cos(x)  |    /    pi\
|----------- - -----------|*cos|x - --|
| 1 - cos(x)      /    pi\|    \    2 /
|              cos|x - --||            
\                 \    2 //
(−cos⁡(x)cos⁡(x−π2)+cos⁡(x−π2)1−cos⁡(x))cos⁡(x−π2)\left(- \frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{1 - \cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)} 
/  sin(x)      sin(2*x)\       
|---------- - ---------|*sin(x)
|1 - cos(x)        2   |       
\             2*sin (x)/
(−sin⁡(2x)2sin⁡2(x)+sin⁡(x)1−cos⁡(x))sin⁡(x)\left(- \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) \sin{\left(x \right)} 
  /                           /x\            \       
  |                      2*cot|-|            |       
  |                           \2/            |    /x\
2*|-cot(x) + --------------------------------|*cot|-|
  |                        /            2/x\\|    \2/
  |                        |    -1 + cot |-|||       
  |          /       2/x\\ |             \2/||       
  |          |1 + cot |-||*|1 - ------------||       
  |          \        \2// |           2/x\ ||       
  |                        |    1 + cot |-| ||       
  \                        \            \2/ //       
-----------------------------------------------------
                            2/x\                     
                     1 + cot |-|                     
                             \2/
2(−cot⁡(x)+2cot⁡(x2)(−cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1))cot⁡(x2)cot⁡2(x2)+1\frac{2 \left(- \cot{\left(x \right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right) \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} 
    1                  /x\
---------- - cot(x)*cot|-|
1 - cos(x)             \2/
−cot⁡(x2)cot⁡(x)+11−cos⁡(x)- \cot{\left(\frac{x}{2} \right)} \cot{\left(x \right)} + \frac{1}{1 - \cos{\left(x \right)}} 
                            /x\     
                  cos(x)*cos|-|     
    1                       \2/     
---------- - -----------------------
1 - cos(x)      /    pi\    /x   pi\
             cos|x - --|*cos|- - --|
                \    2 /    \2   2 /
−cos⁡(x2)cos⁡(x)cos⁡(x2−π2)cos⁡(x−π2)+11−cos⁡(x)- \frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{1 - \cos{\left(x \right)}} 
/     sin(x)        sin(2*x)\       
|--------------- - ---------|*sin(x)
|       /    pi\        2   |       
|1 - sin|x + --|   2*sin (x)|       
\       \    2 /            /
(−sin⁡(2x)2sin⁡2(x)+sin⁡(x)1−sin⁡(x+π2))sin⁡(x)\left(- \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{1 - \sin{\left(x + \frac{\pi}{2} \right)}}\right) \sin{\left(x \right)} 
    1              1      
---------- - -------------
1 - cos(x)             /x\
             tan(x)*tan|-|
                       \2/
−1tan⁡(x2)tan⁡(x)+11−cos⁡(x)- \frac{1}{\tan{\left(\frac{x}{2} \right)} \tan{\left(x \right)}} + \frac{1}{1 - \cos{\left(x \right)}} 
         1            csc(x)
------------------- - ------
/      1   \          sec(x)
|1 - ------|*csc(x)         
\    sec(x)/                
----------------------------
           csc(x)
−csc⁡(x)sec⁡(x)+1(1−1sec⁡(x))csc⁡(x)csc⁡(x)\frac{- \frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}}{\csc{\left(x \right)}} 
       1                1      
--------------- - -------------
           2/x\             /x\
    1 - tan |-|   tan(x)*tan|-|
            \2/             \2/
1 - -----------                
           2/x\                
    1 + tan |-|                
            \2/
−1tan⁡(x2)tan⁡(x)+1−1−tan⁡2(x2)tan⁡2(x2)+1+1- \frac{1}{\tan{\left(\frac{x}{2} \right)} \tan{\left(x \right)}} + \frac{1}{- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1} 
           1                  csc(x)  
------------------------ - -----------
/         1     \             /pi    \
|1 - -----------|*csc(x)   csc|-- - x|
|       /pi    \|             \2     /
|    csc|-- - x||                     
\       \2     //                     
--------------------------------------
                csc(x)
−csc⁡(x)csc⁡(−x+π2)+1(1−1csc⁡(−x+π2))csc⁡(x)csc⁡(x)\frac{- \frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}}{\csc{\left(x \right)}} 
/  sin(x)     cos(x)\       
|---------- - ------|*sin(x)
\1 - cos(x)   sin(x)/
(−cos⁡(x)sin⁡(x)+sin⁡(x)1−cos⁡(x))sin⁡(x)\left(- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) \sin{\left(x \right)} 
                /    pi\    /x   pi\
             sec|x - --|*sec|- - --|
    1           \    2 /    \2   2 /
---------- - -----------------------
      1                     /x\     
1 - ------        sec(x)*sec|-|     
    sec(x)                  \2/
−sec⁡(x2−π2)sec⁡(x−π2)sec⁡(x2)sec⁡(x)+11−1sec⁡(x)- \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(\frac{x}{2} \right)} \sec{\left(x \right)}} + \frac{1}{1 - \frac{1}{\sec{\left(x \right)}}} 
                                 /x\     
                       csc(x)*csc|-|     
       1                         \2/     
--------------- - -----------------------
         1           /pi    \    /pi   x\
1 - -----------   csc|-- - x|*csc|-- - -|
       /pi    \      \2     /    \2    2/
    csc|-- - x|                          
       \2     /
−csc⁡(x2)csc⁡(x)csc⁡(−x+π2)csc⁡(−x2+π2)+11−1csc⁡(−x+π2)- \frac{\csc{\left(\frac{x}{2} \right)} \csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} 
  /                            /x\           \       
  |                       2*tan|-|           |       
  |    1                       \2/           |    /x\
2*|- ------ + -------------------------------|*tan|-|
  |  tan(x)                 /           2/x\\|    \2/
  |                         |    1 - tan |-|||       
  |           /       2/x\\ |            \2/||       
  |           |1 + tan |-||*|1 - -----------||       
  |           \        \2// |           2/x\||       
  |                         |    1 + tan |-|||       
  \                         \            \2///       
-----------------------------------------------------
                            2/x\                     
                     1 + tan |-|                     
                             \2/
2(−1tan⁡(x)+2tan⁡(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1))tan⁡(x2)tan⁡2(x2)+1\frac{2 \left(- \frac{1}{\tan{\left(x \right)}} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right) \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} 
2*(-1/tan(x) + 2*tan(x/2)/((1 + tan(x/2)^2)*(1 - (1 - tan(x/2)^2)/(1 + tan(x/2)^2))))*tan(x/2)/(1 + tan(x/2)^2)
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