Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2)*cot(x-pi/2)-cos(x+pi/2))
¿Cómo vas a descomponer esta cos(x-pi/2)^(2)*cot(x+pi/2)^(2)/(cos(x+pi/2)+cot(x-pi/2)*cot(x-pi/2)-cos(x+pi/2)) expresión en fracciones?
Solución
Ha introducido
[src]
2/ pi\ 2/ pi\
cos |x - --|*cot |x + --|
\ 2 / \ 2 /
---------------------------------------------------
/ pi\ / pi\ / pi\ / pi\
cos|x + --| + cot|x - --|*cot|x - --| - cos|x + --|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\left(\cos{\left(x + \frac{\pi}{2} \right)} + \cot{\left(x - \frac{\pi}{2} \right)} \cot{\left(x - \frac{\pi}{2} \right)}\right) - \cos{\left(x + \frac{\pi}{2} \right)}}$$
(cos(x - pi/2)^2*cot(x + pi/2)^2)/(cos(x + pi/2) + cot(x - pi/2)*cot(x - pi/2) - cos(x + pi/2))
Respuesta numérica
[src]
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
Potencias
[src]
2 / / pi\ /pi \\ | I*|x - --| I*|-- - x|| | \ 2 / \2 /| |e e | |----------- + -----------| \ 2 2 /
$$\left(\frac{e^{i \left(- x + \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(x - \frac{\pi}{2}\right)}}{2}\right)^{2}$$
2 sin (x)
$$\sin^{2}{\left(x \right)}$$
sin(x)^2
Compilar la expresión
[src]
2/ pi\ 2/ pi\
cos |x - --|*cot |x + --|
\ 2 / \ 2 /
-------------------------
2/ pi\
cot |x - --|
\ 2 /
$$\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\cot^{2}{\left(x - \frac{\pi}{2} \right)}}$$
cos(x - pi/2)^2*cot(x + pi/2)^2/cot(x - pi/2)^2
Parte trigonométrica
[src]
2
/ 2/x pi\\
|1 - cot |- + --||
\ \2 4 //
-------------------
2
/ 2/x pi\\
|1 + cot |- + --||
\ \2 4 //
$$\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
4
/ 2/x\\ 8/x\
2*(1 + cos(x)) - 4*|1 - tan |-|| *cos |-|
\ \4// \4/
$$- 4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{4} \right)} + 2 \left(\cos{\left(x \right)} + 1\right)$$
2
/ 1 \ 2
|1 - ------------| *(1 + sin(x))
| 2/x pi\|
| tan |- + --||
\ \2 4 //
---------------------------------
4
$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4}$$
1
------------
2/pi \
sec |-- - x|
\2 /
$$\frac{1}{\sec^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
2
/ /x\ \
2 | 2*cot|-| |
/ 2/x pi\\ | \2/ |
|1 - cot |- + --|| *|1 + -----------|
\ \2 4 // | 2/x\|
| 1 + cot |-||
\ \2//
--------------------------------------
4
$$\frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2}}{4}$$
2
/ /x\ \
2 | 2*tan|-| |
/ 1 \ | \2/ |
|1 - ------------| *|1 + -----------|
| 2/x pi\| | 2/x\|
| tan |- + --|| | 1 + tan |-||
\ \2 4 // \ \2//
--------------------------------------
4
$$\frac{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{4}$$
4 4
- ------------ + ------------
4/pi x\ 2/pi x\
csc |-- - -| csc |-- - -|
\2 2/ \2 2/
$$\frac{4}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{4}{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
4 2
/ 2/x\\ / 2/x\\
4*|-1 + cot |-|| 4*|-1 + cot |-||
\ \4// \ \4//
- ----------------- + -----------------
4 2
/ 2/x\\ / 2/x\\
|1 + cot |-|| |1 + cot |-||
\ \4// \ \4//
$$- \frac{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
2/x\
4*cot |-|
\2/
--------------
2
/ 2/x\\
|1 + cot |-||
\ \2//
$$\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
1
------------
2/ pi\
sec |x - --|
\ 2 /
$$\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$
4 2
/ 2/x\\ / 2/x\\
4*|1 - tan |-|| 4*|1 - tan |-||
\ \4// \ \4//
- ---------------- + ----------------
4 2
/ 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-||
\ \4// \ \4//
$$- \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
4
/ 2/x\\ 8/x\
2*(1 + cos(x)) - 4*|-1 + cot |-|| *sin |-|
\ \4// \4/
$$2 \left(\cos{\left(x \right)} + 1\right) - 4 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}$$
4 4
- ------- + -------
4/x\ 2/x\
sec |-| sec |-|
\2/ \2/
$$\frac{4}{\sec^{2}{\left(\frac{x}{2} \right)}} - \frac{4}{\sec^{4}{\left(\frac{x}{2} \right)}}$$
1 cos(2*x) - - -------- 2 2
$$\frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$$
2/ pi\
cos |x - --|
\ 2 /
$$\cos^{2}{\left(x - \frac{\pi}{2} \right)}$$
4/x\ 2/x\
- 4*cos |-| + 4*cos |-|
\2/ \2/
$$- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)}$$
2
/ 1 \
|1 - ------------|
| 2/x pi\|
| tan |- + --||
\ \2 4 //
-------------------
2
/ 1 \
|1 + ------------|
| 2/x pi\|
| tan |- + --||
\ \2 4 //
$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}$$
2
/ 2/x pi\\
|-1 + tan |- + --||
\ \2 4 //
--------------------
2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/x pi\\
| cos |- + --|| 2
| \2 4 /| / / pi\\
|1 - ------------| *|1 + cos|x - --||
| 2/x pi\| \ \ 2 //
| cos |- - --||
\ \2 4 //
--------------------------------------
4
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}}{4}$$
2
/ 2/x pi\\ 2
|1 - cot |- + --|| *(1 + sin(x))
\ \2 4 //
---------------------------------
4
$$\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4}$$
2
/ 2 \
| cos (x) | 2
|1 - --------------| *(1 + sin(x))
| 4/x pi\|
| 4*sin |- + --||
\ \2 4 //
-----------------------------------
4
$$\frac{\left(1 - \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2}}{4}$$
2 sin (x)
$$\sin^{2}{\left(x \right)}$$
2
/ 2/x pi\ \
2 | csc |- + --| |
/ 1 \ | \2 4 / |
|1 + ------| *|1 - --------------|
\ csc(x)/ | 2/ x pi\|
| csc |- - + --||
\ \ 2 4 //
-----------------------------------
4
$$\frac{\left(1 - \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}}{4}$$
2
/ 2/x pi\\
2 | sec |- - --||
/ 1 \ | \2 4 /|
|1 + -----------| *|1 - ------------|
| / pi\| | 2/x pi\|
| sec|x - --|| | sec |- + --||
\ \ 2 // \ \2 4 //
--------------------------------------
4
$$\frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \left(- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2}}{4}$$
1 ------- 2 csc (x)
$$\frac{1}{\csc^{2}{\left(x \right)}}$$
4/pi x\ 2/pi x\
- 4*sin |-- + -| + 4*sin |-- + -|
\2 2/ \2 2/
$$- 4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 4 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}$$
2/x\
4*tan |-|
\2/
--------------
2
/ 2/x\\
|1 + tan |-||
\ \2//
$$\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2
/ 1 \
|-1 + ------------|
| 2/x pi\|
| cot |- + --||
\ \2 4 //
--------------------
2
/ 1 \
|1 + ------------|
| 2/x pi\|
| cot |- + --||
\ \2 4 //
$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2}}$$
2 2
cos (x - pi)*sin (x)
--------------------
2
cos (x)
$$\frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x - \pi \right)}}{\cos^{2}{\left(x \right)}}$$
cos(x - pi)^2*sin(x)^2/cos(x)^2
Unión de expresiones racionales
[src]
2/-pi + 2*x\ 2/pi + 2*x\
cos |---------|*cot |--------|
\ 2 / \ 2 /
------------------------------
2/-pi + 2*x\
cot |---------|
\ 2 /
$$\frac{\cos^{2}{\left(\frac{2 x - \pi}{2} \right)} \cot^{2}{\left(\frac{2 x + \pi}{2} \right)}}{\cot^{2}{\left(\frac{2 x - \pi}{2} \right)}}$$
cos((-pi + 2*x)/2)^2*cot((pi + 2*x)/2)^2/cot((-pi + 2*x)/2)^2
Denominador racional
[src]
2/ pi\ 2
cot |x + --|*sin (x)
\ 2 /
--------------------
2
tan (x)
$$\frac{\sin^{2}{\left(x \right)} \cot^{2}{\left(x + \frac{\pi}{2} \right)}}{\tan^{2}{\left(x \right)}}$$
cot(x + pi/2)^2*sin(x)^2/tan(x)^2