Sr Examen
¿Cómo vas a descomponer esta sin(x+y)/((sin(x)*sin(y))) expresión en fracciones?
Solución
Ha introducido
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sin(x + y) ------------- sin(x)*sin(y)
$$\frac{\sin{\left(x + y \right)}}{\sin{\left(x \right)} \sin{\left(y \right)}}$$
sin(x + y)/((sin(x)*sin(y)))
Simplificación general
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1 1 ------ + ------ tan(x) tan(y)
$$\frac{1}{\tan{\left(y \right)}} + \frac{1}{\tan{\left(x \right)}}$$
1/tan(x) + 1/tan(y)
Potencias
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/ I*(-x - y) I*(x + y)\ 2*I*\- e + e / --------------------------------- / -I*x I*x\ / -I*y I*y\ \- e + e /*\- e + e /
$$\frac{2 i \left(- e^{i \left(- x - y\right)} + e^{i \left(x + y\right)}\right)}{\left(e^{i x} - e^{- i x}\right) \left(e^{i y} - e^{- i y}\right)}$$
2*i*(-exp(i*(-x - y)) + exp(i*(x + y)))/((-exp(-i*x) + exp(i*x))*(-exp(-i*y) + exp(i*y)))
Parte trigonométrica
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/ pi\ / pi\ sec|x - --| sec|y - --| \ 2 / \ 2 / ----------- + ----------- sec(x) sec(y)
$$\frac{\sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}$$
cot(x) + cot(y)
$$\cot{\left(x \right)} + \cot{\left(y \right)}$$
/x\ /y\ /x pi\ /y pi\
cos|-| cos|-| cos|- - --| cos|- - --|
\2/ \2/ \2 2 / \2 2 /
------------- + ------------- - ----------- - -----------
/x pi\ /y pi\ /x\ /y\
2*cos|- - --| 2*cos|- - --| 2*cos|-| 2*cos|-|
\2 2 / \2 2 / \2/ \2/
$$\frac{\cos{\left(\frac{x}{2} \right)}}{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{y}{2} \right)}}{2 \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{y}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{x}{2} \right)}}$$
csc(x)*csc(y) ------------- csc(x + y)
$$\frac{\csc{\left(x \right)} \csc{\left(y \right)}}{\csc{\left(x + y \right)}}$$
2 2
/ 2/x\\ / 2/y\\
|1 + tan |-|| |1 + tan |-||
1 1 \ \4// \ \4//
------ + ------ - ---------------------- - ----------------------
/x\ /y\ / 2/x\\ /x\ / 2/y\\ /y\
tan|-| tan|-| 4*|1 - tan |-||*tan|-| 4*|1 - tan |-||*tan|-|
\2/ \2/ \ \4// \4/ \ \4// \4/
$$\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{\left(\tan^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2}}{4 \left(1 - \tan^{2}{\left(\frac{y}{4} \right)}\right) \tan{\left(\frac{y}{4} \right)}} - \frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \tan{\left(\frac{x}{4} \right)}}$$
1 1 1 1 ------ + ------ - ------ - ------ /x\ /y\ sin(x) sin(y) tan|-| tan|-| \2/ \2/
$$\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}}$$
2 2
/ 2/x\\ / 2/y\\
|1 + cot |-|| |1 + cot |-||
\ \4// \ \4// /x\ /y\
- ----------------------- - ----------------------- + cot|-| + cot|-|
/ 2/x\\ /x\ / 2/y\\ /y\ \2/ \2/
4*|-1 + cot |-||*cot|-| 4*|-1 + cot |-||*cot|-|
\ \4// \4/ \ \4// \4/
$$\cot{\left(\frac{x}{2} \right)} + \cot{\left(\frac{y}{2} \right)} - \frac{\left(\cot^{2}{\left(\frac{y}{4} \right)} + 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{y}{4} \right)} - 1\right) \cot{\left(\frac{y}{4} \right)}} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \cot{\left(\frac{x}{4} \right)}}$$
/x\ /y\ /pi x\ /pi y\
csc|-| csc|-| csc|-- - -| csc|-- - -|
\2/ \2/ \2 2/ \2 2/
------------- + ------------- - ----------- - -----------
/pi x\ /pi y\ /x\ /y\
2*csc|-- - -| 2*csc|-- - -| 2*csc|-| 2*csc|-|
\2 2/ \2 2/ \2/ \2/
$$\frac{\csc{\left(\frac{x}{2} \right)}}{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{\csc{\left(\frac{y}{2} \right)}}{2 \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{y}{2} \right)}} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{x}{2} \right)}}$$
cos(x) cos(y) ----------- + ----------- / pi\ / pi\ cos|x - --| cos|y - --| \ 2 / \ 2 /
$$\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(y \right)}}{\cos{\left(y - \frac{\pi}{2} \right)}}$$
sin(x) sin(y) 1 1
--------- + --------- - -------------------- - --------------------
2/x\ 2/y\ /x\ /pi x\ /y\ /pi y\
2*sin |-| 2*sin |-| 2*sin|-|*sin|-- + -| 2*sin|-|*sin|-- + -|
\2/ \2/ \2/ \2 2/ \2/ \2 2/
$$- \frac{1}{2 \sin{\left(\frac{y}{2} \right)} \sin{\left(\frac{y}{2} + \frac{\pi}{2} \right)}} + \frac{\sin{\left(y \right)}}{2 \sin^{2}{\left(\frac{y}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{x}{2} \right)} \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{\sin{\left(x \right)}}{2 \sin^{2}{\left(\frac{x}{2} \right)}}$$
/x pi\ /y pi\ /x\ /y\
sec|- - --| sec|- - --| sec|-| sec|-|
\2 2 / \2 2 / \2/ \2/
----------- + ----------- - ------------- - -------------
/x\ /y\ /x pi\ /y pi\
2*sec|-| 2*sec|-| 2*sec|- - --| 2*sec|- - --|
\2/ \2/ \2 2 / \2 2 /
$$- \frac{\sec{\left(\frac{x}{2} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\sec{\left(\frac{y}{2} \right)}}{2 \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{y}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{x}{2} \right)}}$$
/ pi\ / pi\
sec|x - --|*sec|y - --|
\ 2 / \ 2 /
-----------------------
/ pi\
sec|x + y - --|
\ 2 /
$$\frac{\sec{\left(x - \frac{\pi}{2} \right)} \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(x + y - \frac{\pi}{2} \right)}}$$
/ pi\
cos|x + y - --|
\ 2 /
-----------------------
/ pi\ / pi\
cos|x - --|*cos|y - --|
\ 2 / \ 2 /
$$\frac{\cos{\left(x + y - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} \cos{\left(y - \frac{\pi}{2} \right)}}$$
1 1 sin(x) sin(y) - ------ - ------ + ---------- + ---------- sin(x) sin(y) 1 - cos(x) 1 - cos(y)
$$- \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}} + \frac{\sin{\left(y \right)}}{1 - \cos{\left(y \right)}} + \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
sin(2*x) sin(2*y)
--------- + ---------
2 2
2*sin (x) 2*sin (y)
$$\frac{\sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}$$
1 1 1 1 ------ + ------ - --------------- - --------------- /x\ /y\ /x\ /x\ /y\ /y\ tan|-| tan|-| 2*cos|-|*sin|-| 2*cos|-|*sin|-| \2/ \2/ \2/ \2/ \2/ \2/
$$\frac{1}{\tan{\left(\frac{y}{2} \right)}} + \frac{1}{\tan{\left(\frac{x}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{y}{2} \right)} \cos{\left(\frac{y}{2} \right)}} - \frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}$$
1 1 ------ + ------ tan(x) tan(y)
$$\frac{1}{\tan{\left(y \right)}} + \frac{1}{\tan{\left(x \right)}}$$
/ 2/x\\ / 2/y\\ /x y\
|1 + cot |-||*|1 + cot |-||*cot|- + -|
\ \2// \ \2// \2 2/
--------------------------------------
/ 2/x y\\ /x\ /y\
2*|1 + cot |- + -||*cot|-|*cot|-|
\ \2 2// \2/ \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{y}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{y}{2} \right)}}$$
/ 2/x\\ / 2/y\\ /x y\
|1 + tan |-||*|1 + tan |-||*tan|- + -|
\ \2// \ \2// \2 2/
--------------------------------------
/ 2/x y\\ /x\ /y\
2*|1 + tan |- + -||*tan|-|*tan|-|
\ \2 2// \2/ \2/
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{y}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} + \frac{y}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{y}{2} \right)}}$$
2/x\ 2/y\
sin |-| sin |-|
sin(x) sin(y) \2/ \2/
-------------- + -------------- - ------- - -------
2*(1 - cos(x)) 2*(1 - cos(y)) sin(x) sin(y)
$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin^{2}{\left(\frac{y}{2} \right)}}{\sin{\left(y \right)}} + \frac{\sin{\left(y \right)}}{2 \left(1 - \cos{\left(y \right)}\right)} + \frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)}$$
/x pi\ /y pi\ /x\ /x pi\ /y\ /y pi\
sec|- - --| sec|- - --| sec|-|*sec|- - --| sec|-|*sec|- - --|
\2 2 / \2 2 / \2/ \2 2 / \2/ \2 2 /
----------- + ----------- - ------------------ - ------------------
/x\ /y\ 2 2
sec|-| sec|-|
\2/ \2/
$$- \frac{\sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2} - \frac{\sec{\left(\frac{y}{2} \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{2} + \frac{\sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{y}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec{\left(\frac{x}{2} \right)}}$$
/x\ /y\ /x\ /pi x\ /y\ /pi y\
csc|-| csc|-| csc|-|*csc|-- - -| csc|-|*csc|-- - -|
\2/ \2/ \2/ \2 2/ \2/ \2 2/
----------- + ----------- - ------------------ - ------------------
/pi x\ /pi y\ 2 2
csc|-- - -| csc|-- - -|
\2 2/ \2 2/
$$- \frac{\csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\csc{\left(\frac{x}{2} \right)}}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(\frac{y}{2} \right)} \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\csc{\left(\frac{y}{2} \right)}}{\csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}$$
1 1 /x\ /y\ - ------ - ------ + cot|-| + cot|-| sin(x) sin(y) \2/ \2/
$$\cot{\left(\frac{x}{2} \right)} + \cot{\left(\frac{y}{2} \right)} - \frac{1}{\sin{\left(y \right)}} - \frac{1}{\sin{\left(x \right)}}$$
2/x\ 2/y\
sin |-| sin |-|
\2/ \2/ sin(x) sin(y)
- ------- - ------- + --------- + ---------
sin(x) sin(y) 2/x\ 2/y\
4*sin |-| 4*sin |-|
\2/ \2/
$$- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - \frac{\sin^{2}{\left(\frac{y}{2} \right)}}{\sin{\left(y \right)}} + \frac{\sin{\left(y \right)}}{4 \sin^{2}{\left(\frac{y}{2} \right)}} + \frac{\sin{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}}$$
/x\ /y\
cos|-| cos|-|
\2/ \2/ 1 1
----------- + ----------- - -------------------- - --------------------
/x pi\ /y pi\ /x\ /x pi\ /y\ /y pi\
cos|- - --| cos|- - --| 2*cos|-|*cos|- - --| 2*cos|-|*cos|- - --|
\2 2 / \2 2 / \2/ \2 2 / \2/ \2 2 /
$$\frac{\cos{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{y}{2} \right)}}{\cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{1}{2 \cos{\left(\frac{y}{2} \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} - \frac{1}{2 \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
csc(x) csc(y) ----------- + ----------- /pi \ /pi \ csc|-- - x| csc|-- - y| \2 / \2 /
$$\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(y \right)}}{\csc{\left(- y + \frac{\pi}{2} \right)}}$$
/x\ /y\
tan|-| tan|-|
1 1 \2/ \2/
-------- + -------- - ------ - ------
/x\ /y\ 2 2
2*tan|-| 2*tan|-|
\2/ \2/
$$- \frac{\tan{\left(\frac{x}{2} \right)}}{2} - \frac{\tan{\left(\frac{y}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{y}{2} \right)}} + \frac{1}{2 \tan{\left(\frac{x}{2} \right)}}$$
/x\ /y\
cot|-| cot|-|
\2/ \2/ 1 1
------ + ------ - -------- - --------
2 2 /x\ /y\
2*cot|-| 2*cot|-|
\2/ \2/
$$\frac{\cot{\left(\frac{x}{2} \right)}}{2} + \frac{\cot{\left(\frac{y}{2} \right)}}{2} - \frac{1}{2 \cot{\left(\frac{y}{2} \right)}} - \frac{1}{2 \cot{\left(\frac{x}{2} \right)}}$$
cot(x/2)/2 + cot(y/2)/2 - 1/(2*cot(x/2)) - 1/(2*cot(y/2))