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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    /   /x   pi\      /x   pi\\
    |sin|- - --| + cos|- - --||
    |   \2   2 /      \2   2 /|
atan|-------------------------|
    |   /x   pi\      /x   pi\|
    |sin|- + --| + cos|- + --||
    \   \2   2 /      \2   2 //
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + \cos{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} \right)}$$ 
atan((sin(x/2 - pi/2) + cos(x/2 - pi/2))/(sin(x/2 + pi/2) + cos(x/2 + pi/2)))
Descomposición de una fracción [src] 
atan(sin(x/2)/(-sin(x/2) + cos(x/2)) - cos(x/2)/(-sin(x/2) + cos(x/2)))
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
    /         /x\                 /x\     \
    |      sin|-|              cos|-|     |
    |         \2/                 \2/     |
atan|----------------- - -----------------|
    |     /x\      /x\        /x\      /x\|
    |- sin|-| + cos|-|   - sin|-| + cos|-||
    \     \2/      \2/        \2/      \2//
Simplificación general [src] 
-pi 
----
 4
$$- \frac{\pi}{4}$$ 
-pi/4
Respuesta numérica [src] 
atan((sin(x/2 - pi/2) + cos(x/2 - pi/2))/(sin(x/2 + pi/2) + cos(x/2 + pi/2)))
atan((sin(x/2 - pi/2) + cos(x/2 - pi/2))/(sin(x/2 + pi/2) + cos(x/2 + pi/2)))
Combinatoria [src] 
    /         /x\                 /x\     \
    |      sin|-|              cos|-|     |
    |         \2/                 \2/     |
atan|----------------- - -----------------|
    |     /x\      /x\        /x\      /x\|
    |- sin|-| + cos|-|   - sin|-| + cos|-||
    \     \2/      \2/        \2/      \2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
atan(sin(x/2)/(-sin(x/2) + cos(x/2)) - cos(x/2)/(-sin(x/2) + cos(x/2)))
Potencias [src] 
    /     /pi   x\      /x   pi\     /     /pi   x\      /x   pi\\  \
    |   I*|-- - -|    I*|- - --|     |   I*|-- - -|    I*|- - --||  |
    |     \2    2/      \2   2 /     |     \2    2/      \2   2 /|  |
    |  e             e             I*\- e           + e          /  |
    |  ----------- + ----------- - -------------------------------  |
    |       2             2                       2                 |
atan|---------------------------------------------------------------|
    |   /pi   x\      /  pi   x\     /     /  pi   x\      /pi   x\\|
    | I*|-- + -|    I*|- -- - -|     |   I*|- -- - -|    I*|-- + -|||
    |   \2    2/      \  2    2/     |     \  2    2/      \2    2/||
    |e             e               I*\- e             + e          /|
    |----------- + ------------- - ---------------------------------|
    \     2              2                         2                /
$$\operatorname{atan}{\left(\frac{- \frac{i \left(- e^{i \left(- \frac{x}{2} + \frac{\pi}{2}\right)} + e^{i \left(\frac{x}{2} - \frac{\pi}{2}\right)}\right)}{2} + \frac{e^{i \left(- \frac{x}{2} + \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(\frac{x}{2} - \frac{\pi}{2}\right)}}{2}}{- \frac{i \left(- e^{i \left(- \frac{x}{2} - \frac{\pi}{2}\right)} + e^{i \left(\frac{x}{2} + \frac{\pi}{2}\right)}\right)}{2} + \frac{e^{i \left(- \frac{x}{2} - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(\frac{x}{2} + \frac{\pi}{2}\right)}}{2}} \right)}$$ 
    /     /x\      /x\\
    |- cos|-| + sin|-||
    |     \2/      \2/|
atan|-----------------|
    |     /x\      /x\|
    |- sin|-| + cos|-||
    \     \2/      \2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
-pi 
----
 4
$$- \frac{\pi}{4}$$ 
-pi/4
Denominador común [src] 
    /         /x\                 /x\     \
    |      sin|-|              cos|-|     |
    |         \2/                 \2/     |
atan|----------------- - -----------------|
    |     /x\      /x\        /x\      /x\|
    |- sin|-| + cos|-|   - sin|-| + cos|-||
    \     \2/      \2/        \2/      \2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
atan(sin(x/2)/(-sin(x/2) + cos(x/2)) - cos(x/2)/(-sin(x/2) + cos(x/2)))
Denominador racional [src] 
    /         /x\      /x\    \
    |    - cos|-| + sin|-|    |
    |         \2/      \2/    |
atan|-------------------------|
    |   /x   pi\      /x   pi\|
    |sin|- + --| + cos|- + --||
    \   \2   2 /      \2   2 //
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + \cos{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} \right)}$$ 
atan((-cos(x/2) + sin(x/2))/(sin(x/2 + pi/2) + cos(x/2 + pi/2)))
Unión de expresiones racionales [src] 
    /   /x - pi\      /x - pi\\
    |cos|------| + sin|------||
    |   \  2   /      \  2   /|
atan|-------------------------|
    |   /pi + x\      /pi + x\|
    |cos|------| + sin|------||
    \   \  2   /      \  2   //
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x - \pi}{2} \right)} + \cos{\left(\frac{x - \pi}{2} \right)}}{\sin{\left(\frac{x + \pi}{2} \right)} + \cos{\left(\frac{x + \pi}{2} \right)}} \right)}$$ 
atan((cos((x - pi)/2) + sin((x - pi)/2))/(cos((pi + x)/2) + sin((pi + x)/2)))
Abrimos la expresión [src] 
    /         /x\                 /x\     \
    |      sin|-|              cos|-|     |
    |         \2/                 \2/     |
atan|----------------- - -----------------|
    |     /x\      /x\        /x\      /x\|
    |- sin|-| + cos|-|   - sin|-| + cos|-||
    \     \2/      \2/        \2/      \2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
atan(sin(x/2)/(-sin(x/2) + cos(x/2)) - cos(x/2)/(-sin(x/2) + cos(x/2)))
Parte trigonométrica [src] 
    /   /x     \      /x\\
    |cos|- - pi| + sin|-||
    |   \2     /      \2/|
atan|--------------------|
    |      /x\      /x\  |
    | - sin|-| + cos|-|  |
    \      \2/      \2/  /
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} - \pi \right)}}{- \sin{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
    /  1           1     \
    |------ + -----------|
    |   /x\      /x     \|
    |csc|-|   sec|- - pi||
    |   \2/      \2     /|
atan|--------------------|
    |    1        1      |
    |  ------ - ------   |
    |     /x\      /x\   |
    |  sec|-|   csc|-|   |
    \     \2/      \2/   /
$$\operatorname{atan}{\left(\frac{\frac{1}{\sec{\left(\frac{x}{2} - \pi \right)}} + \frac{1}{\csc{\left(\frac{x}{2} \right)}}}{\frac{1}{\sec{\left(\frac{x}{2} \right)}} - \frac{1}{\csc{\left(\frac{x}{2} \right)}}} \right)}$$ 
    /          1                                       \
    |-1 + ------------                                 |
    |        2/pi   x\                                 |
    |     cot |-- + -|                                 |
    |         \4    4/                 2               |
    |----------------- - ------------------------------|
    |          1         /         1      \    /pi   x\|
    | 1 + ------------   |1 + ------------|*cot|-- + -||
    |        2/pi   x\   |       2/pi   x\|    \4    4/|
    |     cot |-- + -|   |    cot |-- + -||            |
    |         \4    4/   \        \4    4//            |
atan|--------------------------------------------------|
    |               2/pi   x\         /pi   x\         |
    |       -1 + cot |-- + -|    2*cot|-- + -|         |
    |                \4    4/         \4    4/         |
    |       ----------------- + ----------------       |
    |               2/pi   x\          2/pi   x\       |
    |        1 + cot |-- + -|   1 + cot |-- + -|       |
    \                \4    4/           \4    4/       /
$$\operatorname{atan}{\left(\frac{\frac{-1 + \frac{1}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}} - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}\right) \cot{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}}{\frac{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}} \right)}$$ 
    /         1                                       \
    |1 - ------------                                 |
    |       2/pi   x\                                 |
    |    tan |-- + -|                                 |
    |        \4    4/                 2               |
    |---------------- - ------------------------------|
    |         1         /         1      \    /pi   x\|
    |1 + ------------   |1 + ------------|*tan|-- + -||
    |       2/pi   x\   |       2/pi   x\|    \4    4/|
    |    tan |-- + -|   |    tan |-- + -||            |
    |        \4    4/   \        \4    4//            |
atan|-------------------------------------------------|
    |              2/pi   x\         /pi   x\         |
    |       1 - tan |-- + -|    2*tan|-- + -|         |
    |               \4    4/         \4    4/         |
    |       ---------------- + ----------------       |
    |              2/pi   x\          2/pi   x\       |
    |       1 + tan |-- + -|   1 + tan |-- + -|       |
    \               \4    4/           \4    4/       /
$$\operatorname{atan}{\left(\frac{\frac{1 - \frac{1}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}} - \frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}\right) \tan{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}}{\frac{1 - \tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}} \right)}$$ 
    /  1        1   \
    |------ - ------|
    |   /x\      /x\|
    |csc|-|   sec|-||
    |   \2/      \2/|
atan|---------------|
    |  1        1   |
    |------ - ------|
    |   /x\      /x\|
    |sec|-|   csc|-||
    \   \2/      \2//
$$\operatorname{atan}{\left(\frac{- \frac{1}{\sec{\left(\frac{x}{2} \right)}} + \frac{1}{\csc{\left(\frac{x}{2} \right)}}}{\frac{1}{\sec{\left(\frac{x}{2} \right)}} - \frac{1}{\csc{\left(\frac{x}{2} \right)}}} \right)}$$ 
    /      /x\      /x\  \
    | - cos|-| + sin|-|  |
    |      \2/      \2/  |
atan|--------------------|
    |   /x\      /     x\|
    |cos|-| + sin|pi + -||
    \   \2/      \     2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} + \pi \right)} + \cos{\left(\frac{x}{2} \right)}} \right)}$$ 
    /     /x\      /x   pi\\
    |- cos|-| + cos|- - --||
    |     \2/      \2   2 /|
atan|----------------------|
    |     /x   pi\      /x\|
    |- cos|- - --| + cos|-||
    \     \2   2 /      \2//
$$\operatorname{atan}{\left(\frac{- \cos{\left(\frac{x}{2} \right)} + \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}$$ 
    /     /pi   x\      /x\\
    |- sin|-- + -| + sin|-||
    |     \2    2/      \2/|
atan|----------------------|
    |     /x\      /pi   x\|
    |- sin|-| + sin|-- + -||
    \     \2/      \2    2//
$$\operatorname{atan}{\left(\frac{\sin{\left(\frac{x}{2} \right)} - \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{- \sin{\left(\frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} \right)}$$ 
    /     1          1   \
    |----------- - ------|
    |   /pi   x\      /x\|
    |sec|-- - -|   sec|-||
    |   \2    2/      \2/|
atan|--------------------|
    |  1           1     |
    |------ - -----------|
    |   /x\      /pi   x\|
    |sec|-|   sec|-- - -||
    \   \2/      \2    2//
$$\operatorname{atan}{\left(\frac{\frac{1}{\sec{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(\frac{x}{2} \right)}}}{- \frac{1}{\sec{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(\frac{x}{2} \right)}}} \right)}$$ 
    /        2/pi   x\         /pi   x\  \
    |-1 + tan |-- + -|    2*tan|-- + -|  |
    |         \4    4/         \4    4/  |
    |----------------- - ----------------|
    |        2/pi   x\          2/pi   x\|
    | 1 + tan |-- + -|   1 + tan |-- + -||
    |         \4    4/           \4    4/|
atan|------------------------------------|
    |        2/pi   x\         /pi   x\  |
    |-1 + cot |-- + -|    2*cot|-- + -|  |
    |         \4    4/         \4    4/  |
    |----------------- + ----------------|
    |        2/pi   x\          2/pi   x\|
    | 1 + cot |-- + -|   1 + cot |-- + -||
    \         \4    4/           \4    4//
$$\operatorname{atan}{\left(\frac{\frac{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} - \frac{2 \tan{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}}{\frac{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}} \right)}$$ 
    /  1           1     \
    |------ - -----------|
    |   /x\      /pi   x\|
    |csc|-|   csc|-- - -||
    |   \2/      \2    2/|
atan|--------------------|
    |     1          1   |
    |----------- - ------|
    |   /pi   x\      /x\|
    |csc|-- - -|   csc|-||
    \   \2    2/      \2//
$$\operatorname{atan}{\left(\frac{- \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(\frac{x}{2} \right)}}}{\frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(\frac{x}{2} \right)}}} \right)}$$ 
-pi 
----
 4
$$- \frac{\pi}{4}$$ 
    /       2/pi   x\         /pi   x\  \
    |1 - cot |-- + -|    2*cot|-- + -|  |
    |        \4    4/         \4    4/  |
    |---------------- - ----------------|
    |       2/pi   x\          2/pi   x\|
    |1 + cot |-- + -|   1 + cot |-- + -||
    |        \4    4/           \4    4/|
atan|-----------------------------------|
    |       2/pi   x\         /pi   x\  |
    |1 - tan |-- + -|    2*tan|-- + -|  |
    |        \4    4/         \4    4/  |
    |---------------- + ----------------|
    |       2/pi   x\          2/pi   x\|
    |1 + tan |-- + -|   1 + tan |-- + -||
    \        \4    4/           \4    4//
$$\operatorname{atan}{\left(\frac{\frac{1 - \cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} - \frac{2 \cot{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}}{\frac{1 - \tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{4} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} + \frac{\pi}{4} \right)} + 1}} \right)}$$ 
atan(((1 - cot(pi/4 + x/4)^2)/(1 + cot(pi/4 + x/4)^2) - 2*cot(pi/4 + x/4)/(1 + cot(pi/4 + x/4)^2))/((1 - tan(pi/4 + x/4)^2)/(1 + tan(pi/4 + x/4)^2) + 2*tan(pi/4 + x/4)/(1 + tan(pi/4 + x/4)^2)))
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