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

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

Expresión a simplificar:

Solución

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