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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(3*pi/(2+x))*cot(pi-x)-sin(pi+x)*cos(pi/(2+x))

¿Cómo vas a descomponer esta cot(3*pi/(2+x))*cot(pi-x)-sin(pi+x)*cos(pi/(2+x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   / 3*pi\                              /  pi \
cot|-----|*cot(pi - x) - sin(pi + x)*cos|-----|
   \2 + x/                              \2 + x/
$$- \sin{\left(x + \pi \right)} \cos{\left(\frac{\pi}{x + 2} \right)} + \cot{\left(\frac{3 \pi}{x + 2} \right)} \cot{\left(\pi - x \right)}$$ 
cot((3*pi)/(2 + x))*cot(pi - x) - sin(pi + x)*cos(pi/(2 + x))
Simplificación general [src] 
   /  pi \                    / 3*pi\
cos|-----|*sin(x) - cot(x)*cot|-----|
   \2 + x/                    \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
cos(pi/(2 + x))*sin(x) - cot(x)*cot(3*pi/(2 + x))
Respuesta numérica [src] 
cot((3*pi)/(2 + x))*cot(pi - x) - cos(pi/(2 + x))*sin(pi + x)
cot((3*pi)/(2 + x))*cot(pi - x) - cos(pi/(2 + x))*sin(pi + x)
Combinatoria [src] 
   /  pi \                    / 3*pi\
cos|-----|*sin(x) - cot(x)*cot|-----|
   \2 + x/                    \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
cos(pi/(2 + x))*sin(x) - cot(x)*cot(3*pi/(2 + x))
Unión de expresiones racionales [src] 
   /  pi \                    / 3*pi\
cos|-----|*sin(x) - cot(x)*cot|-----|
   \2 + x/                    \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
cos(pi/(2 + x))*sin(x) - cot(x)*cot(3*pi/(2 + x))
Denominador racional [src] 
   /  pi \                    / 3*pi\
cos|-----|*sin(x) - cot(x)*cot|-----|
   \2 + x/                    \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
cos(pi/(2 + x))*sin(x) - cot(x)*cot(3*pi/(2 + x))
Potencias [src] 
   /  pi \                    / 3*pi\
cos|-----|*sin(x) - cot(x)*cot|-----|
   \2 + x/                    \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
                        /  pi*I    -pi*I \                               
                        | -----    ------|                               
                        | 2 + x    2 + x |                               
                        |e        e      | /   I*(-pi - x)    I*(pi + x)\
                      I*|------ + -------|*\- e            + e          /
            / 3*pi\     \  2         2   /                               
- cot(x)*cot|-----| + ---------------------------------------------------
            \2 + x/                            2
$$\frac{i \left(- e^{i \left(- x - \pi\right)} + e^{i \left(x + \pi\right)}\right) \left(\frac{e^{\frac{i \pi}{x + 2}}}{2} + \frac{e^{- \frac{i \pi}{x + 2}}}{2}\right)}{2} - \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}$$ 
-cot(x)*cot(3*pi/(2 + x)) + i*(exp(pi*i/(2 + x))/2 + exp(-pi*i/(2 + x))/2)*(-exp(i*(-pi - x)) + exp(i*(pi + x)))/2
Abrimos la expresión [src] 
                        / 3*pi\                  / 3*pi\
                     cot|-----|    zoo*cot(x)*cot|-----|
   /  pi \              \2 + x/                  \2 + x/
cos|-----|*sin(x) - ------------ + ---------------------
   \2 + x/          zoo - cot(x)        zoo - cot(x)
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} + \frac{\tilde{\infty} \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)}}{- \cot{\left(x \right)} + \tilde{\infty}} - \frac{\cot{\left(\frac{3 \pi}{x + 2} \right)}}{- \cot{\left(x \right)} + \tilde{\infty}}$$ 
cos(pi/(2 + x))*sin(x) - cot((3*pi)/(2 + x))/(±oo - cot(x)) + ±oo*cot(x)*cot((3*pi)/(2 + x))/(±oo - cot(x))
Parte trigonométrica [src] 
                           /       2/    pi   \\    /x\  
                         2*|1 - tan |---------||*cot|-|  
          1                \        \2*(2 + x)//    \2/  
- ----------------- + -----------------------------------
            / 3*pi\   /       2/x\\ /       2/    pi   \\
  tan(x)*tan|-----|   |1 + cot |-||*|1 + tan |---------||
            \2 + x/   \        \2// \        \2*(2 + x)//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)}\right) \cot{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{1}{\tan{\left(x \right)} \tan{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                              /    pi\    /  pi    3*pi\
                           sec|x - --|*sec|- -- + -----|
            1                 \    2 /    \  2    2 + x/
- ---------------------- - -----------------------------
     /  pi \    /    pi\                   / 3*pi\      
  sec|-----|*sec|x + --|         sec(x)*sec|-----|      
     \2 + x/    \    2 /                   \2 + x/
$$- \frac{1}{\sec{\left(\frac{\pi}{x + 2} \right)} \sec{\left(x + \frac{\pi}{2} \right)}} - \frac{\sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{x + 2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)} \sec{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                          /        2/    pi   \\    /x\  
                        2*|-1 + cot |---------||*cot|-|  
            / 3*pi\       \         \2*(2 + x)//    \2/  
- cot(x)*cot|-----| + -----------------------------------
            \2 + x/   /       2/x\\ /       2/    pi   \\
                      |1 + cot |-||*|1 + cot |---------||
                      \        \2// \        \2*(2 + x)//
$$- \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)} + \frac{2 \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right)}$$ 
                           /       2/    pi   \\    /x\  
                         2*|1 - tan |---------||*tan|-|  
          1                \        \2*(2 + x)//    \2/  
- ----------------- + -----------------------------------
            / 3*pi\   /       2/x\\ /       2/    pi   \\
  tan(x)*tan|-----|   |1 + tan |-||*|1 + tan |---------||
            \2 + x/   \        \2// \        \2*(2 + x)//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right)} - \frac{1}{\tan{\left(x \right)} \tan{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                                 /       2/    pi   \\          
                               2*|1 - tan |---------||          
          1                      \        \2*(2 + x)//          
- ----------------- + ------------------------------------------
            / 3*pi\   /       1   \ /       2/    pi   \\    /x\
  tan(x)*tan|-----|   |1 + -------|*|1 + tan |---------||*tan|-|
            \2 + x/   |       2/x\| \        \2*(2 + x)//    \2/
                      |    tan |-||                             
                      \        \2//
$$- \frac{1}{\tan{\left(x \right)} \tan{\left(\frac{3 \pi}{x + 2} \right)}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}$$ 
                                 /        2/    pi   \\         
                               2*|-1 + cot |---------||         
            / 3*pi\              \         \2*(2 + x)//         
- cot(x)*cot|-----| + ------------------------------------------
            \2 + x/   /       1   \ /       2/    pi   \\    /x\
                      |1 + -------|*|1 + cot |---------||*cot|-|
                      |       2/x\| \        \2*(2 + x)//    \2/
                      |    cot |-||                             
                      \        \2//
$$- \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)} + \frac{2 \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} - 1\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}$$ 
                                      / 6*pi\ 
                          sin(2*x)*sin|-----| 
          /pi     pi \                \2 + x/ 
sin(x)*sin|-- + -----| - ---------------------
          \2    2 + x/        2       2/ 3*pi\
                         4*sin (x)*sin |-----|
                                       \2 + x/
$$\sin{\left(x \right)} \sin{\left(\frac{\pi}{2} + \frac{\pi}{x + 2} \right)} - \frac{\sin{\left(2 x \right)} \sin{\left(\frac{6 \pi}{x + 2} \right)}}{4 \sin^{2}{\left(x \right)} \sin^{2}{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                          /        2/    pi   \\    /x\  
                        2*|-1 + cot |---------||*tan|-|  
            / 3*pi\       \         \2*(2 + x)//    \2/  
- cot(x)*cot|-----| + -----------------------------------
            \2 + x/   /       2/    pi   \\ /       2/x\\
                      |1 + cot |---------||*|1 + tan |-||
                      \        \2*(2 + x)// \        \2//
$$- \cot{\left(x \right)} \cot{\left(\frac{3 \pi}{x + 2} \right)} + \frac{2 \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} - 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi}{2 \left(x + 2\right)} \right)} + 1\right)}$$ 
                                        / 3*pi\     
                              csc(x)*csc|-----|     
          1                             \2 + x/     
---------------------- - ---------------------------
          /pi     pi \      /pi    \    /pi    3*pi\
csc(x)*csc|-- - -----|   csc|-- - x|*csc|-- - -----|
          \2    2 + x/      \2     /    \2    2 + x/
$$- \frac{\csc{\left(x \right)} \csc{\left(\frac{3 \pi}{x + 2} \right)}}{\csc{\left(\frac{\pi}{2} - \frac{3 \pi}{x + 2} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)} \csc{\left(\frac{\pi}{2} - \frac{\pi}{x + 2} \right)}}$$ 
                                           / 3*pi\      
                                 cos(x)*cos|-----|      
     /  pi \    /    pi\                   \2 + x/      
- cos|-----|*cos|x + --| - -----------------------------
     \2 + x/    \    2 /      /pi    \    /  pi    3*pi\
                           cos|-- - x|*cos|- -- + -----|
                              \2     /    \  2    2 + x/
$$- \frac{\cos{\left(x \right)} \cos{\left(\frac{3 \pi}{x + 2} \right)}}{\cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{x + 2} \right)} \cos{\left(- x + \frac{\pi}{2} \right)}} - \cos{\left(\frac{\pi}{x + 2} \right)} \cos{\left(x + \frac{\pi}{2} \right)}$$ 
                                         / 3*pi\      
                               cos(x)*cos|-----|      
   /  pi \    /    pi\                   \2 + x/      
cos|-----|*cos|x - --| - -----------------------------
   \2 + x/    \    2 /      /    pi\    /  pi    3*pi\
                         cos|x - --|*cos|- -- + -----|
                            \    2 /    \  2    2 + x/
$$- \frac{\cos{\left(x \right)} \cos{\left(\frac{3 \pi}{x + 2} \right)}}{\cos{\left(- \frac{\pi}{2} + \frac{3 \pi}{x + 2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}} + \cos{\left(\frac{\pi}{x + 2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$ 
                            /    pi\    /  pi    3*pi\
                         sec|x - --|*sec|- -- + -----|
          1                 \    2 /    \  2    2 + x/
---------------------- - -----------------------------
   /  pi \    /    pi\                   / 3*pi\      
sec|-----|*sec|x - --|         sec(x)*sec|-----|      
   \2 + x/    \    2 /                   \2 + x/
$$\frac{1}{\sec{\left(\frac{\pi}{x + 2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\sec{\left(- \frac{\pi}{2} + \frac{3 \pi}{x + 2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)} \sec{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
   /  pi \             / 3*pi\            
cos|-----|*sin(x) + cot|-----|*cot(pi - x)
   \2 + x/             \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} + \cot{\left(\frac{3 \pi}{x + 2} \right)} \cot{\left(\pi - x \right)}$$ 
                                 / 6*pi\ 
                     sin(2*x)*sin|-----| 
   /  pi \                       \2 + x/ 
cos|-----|*sin(x) - ---------------------
   \2 + x/               2       2/ 3*pi\
                    4*sin (x)*sin |-----|
                                  \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \frac{\sin{\left(2 x \right)} \sin{\left(\frac{6 \pi}{x + 2} \right)}}{4 \sin^{2}{\left(x \right)} \sin^{2}{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
   /  pi \                  1        
cos|-----|*sin(x) - -----------------
   \2 + x/                    / 3*pi\
                    tan(x)*tan|-----|
                              \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \frac{1}{\tan{\left(x \right)} \tan{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                              / 3*pi\
                    cos(x)*cos|-----|
   /  pi \                    \2 + x/
cos|-----|*sin(x) - -----------------
   \2 + x/                    / 3*pi\
                    sin(x)*sin|-----|
                              \2 + x/
$$\sin{\left(x \right)} \cos{\left(\frac{\pi}{x + 2} \right)} - \frac{\cos{\left(x \right)} \cos{\left(\frac{3 \pi}{x + 2} \right)}}{\sin{\left(x \right)} \sin{\left(\frac{3 \pi}{x + 2} \right)}}$$ 
                              / 3*pi\
                    csc(x)*csc|-----|
        1                     \2 + x/
----------------- - -----------------
          /  pi \             / 3*pi\
csc(x)*sec|-----|   sec(x)*sec|-----|
          \2 + x/             \2 + x/
$$- \frac{\csc{\left(x \right)} \csc{\left(\frac{3 \pi}{x + 2} \right)}}{\sec{\left(x \right)} \sec{\left(\frac{3 \pi}{x + 2} \right)}} + \frac{1}{\csc{\left(x \right)} \sec{\left(\frac{\pi}{x + 2} \right)}}$$ 
1/(csc(x)*sec(pi/(2 + x))) - csc(x)*csc(3*pi/(2 + x))/(sec(x)*sec(3*pi/(2 + x)))
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