Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)^2*cos(x)/(2-sin(x))^2+2*cos(x)*sin(x)/(2-sin(x))
¿Cómo vas a descomponer esta sin(x)^2*cos(x)/(2-sin(x))^2+2*cos(x)*sin(x)/(2-sin(x)) expresión en fracciones?
Solución
Ha introducido
[src]
2
sin (x)*cos(x) 2*cos(x)*sin(x)
-------------- + ---------------
2 2 - sin(x)
(2 - sin(x))
$$\frac{\sin{\left(x \right)} 2 \cos{\left(x \right)}}{2 - \sin{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
(sin(x)^2*cos(x))/(2 - sin(x))^2 + ((2*cos(x))*sin(x))/(2 - sin(x))
Simplificación general
[src]
(4 - sin(x))*cos(x)*sin(x)
--------------------------
2
(-2 + sin(x))
$$\frac{\left(4 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
(4 - sin(x))*cos(x)*sin(x)/(-2 + sin(x))^2
Respuesta numérica
[src]
0.25*sin(x)^2*cos(x)/(1 - 0.5*sin(x))^2 + 2.0*cos(x)*sin(x)/(2.0 - sin(x))
0.25*sin(x)^2*cos(x)/(1 - 0.5*sin(x))^2 + 2.0*cos(x)*sin(x)/(2.0 - sin(x))
Unión de expresiones racionales
[src]
(4 - sin(x))*cos(x)*sin(x)
--------------------------
2
(2 - sin(x))
$$\frac{\left(4 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
(4 - sin(x))*cos(x)*sin(x)/(2 - sin(x))^2
Combinatoria
[src]
-(-4 + sin(x))*cos(x)*sin(x)
-----------------------------
2
(-2 + sin(x))
$$- \frac{\left(\sin{\left(x \right)} - 4\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
-(-4 + sin(x))*cos(x)*sin(x)/(-2 + sin(x))^2
Denominador racional
[src]
2 2
sin (x)*(2 - sin(x))*cos(x) + 2*(2 - sin(x)) *cos(x)*sin(x)
-----------------------------------------------------------
3
(2 - sin(x))
$$\frac{2 \left(2 - \sin{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \left(2 - \sin{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{3}}$$
(sin(x)^2*(2 - sin(x))*cos(x) + 2*(2 - sin(x))^2*cos(x)*sin(x))/(2 - sin(x))^3
Denominador común
[src]
4*cos(x)
-cos(x) + ----------------------
2
4 + sin (x) - 4*sin(x)
$$- \cos{\left(x \right)} + \frac{4 \cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 4}$$
-cos(x) + 4*cos(x)/(4 + sin(x)^2 - 4*sin(x))
Potencias
[src]
2 / I*x -I*x\
/ -I*x I*x\ |e e |
\- e + e / *|---- + -----| / -I*x I*x\ / I*x -I*x\
\ 2 2 / I*\- e + e /*\e + e /
- -------------------------------- - ---------------------------------
2 / / -I*x I*x\\
/ / -I*x I*x\\ | I*\- e + e /|
| I*\- e + e /| 2*|2 + ------------------|
4*|2 + ------------------| \ 2 /
\ 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right) \left(e^{i x} + e^{- i x}\right)}{2 \left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 2\right)} - \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)^{2}}{4 \left(\frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 2\right)^{2}}$$
2
sin (x)*cos(x) 2*cos(x)*sin(x)
-------------- + ---------------
2 2 - sin(x)
(2 - sin(x))
$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 - \sin{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
sin(x)^2*cos(x)/(2 - sin(x))^2 + 2*cos(x)*sin(x)/(2 - sin(x))
Compilar la expresión
[src]
2
sin (x)*cos(x) 2*cos(x)*sin(x)
-------------- + ---------------
2 2 - sin(x)
(2 - sin(x))
$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 - \sin{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
sin(x)^2*cos(x)/(2 - sin(x))^2 + 2*cos(x)*sin(x)/(2 - sin(x))
Parte trigonométrica
[src]
6 4
/ 2/x\\ / 1 \ / 2/x\\ / 1 \
4*|-1 + cot |-|| *|1 - -------| 4*|-1 + cot |-|| *|1 - -------|
\ \4// | 2/x\| \ \4// | 2/x\|
| cot |-|| | cot |-||
\ \2// \ \2//
------------------------------------------ + ---------------------------------------
2 / /x\ \
/ /x\ \ 4 | 2*cot|-| |
6 | 2*cot|-| | / 2/x\\ | \2/ | /x\
/ 2/x\\ | \2/ | 2/x\ |1 + cot |-|| *|2 - -----------|*cot|-|
|1 + cot |-|| *|-2 + -----------| *cot |-| \ \4// | 2/x\| \2/
\ \4// | 2/x\| \2/ | 1 + cot |-||
| 1 + cot |-|| \ \2//
\ \2//
$$\frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(2 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)}} + \frac{4 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(-2 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \cot^{2}{\left(\frac{x}{2} \right)}}$$
1
4 - -----------
/ pi\
sec|x - --|
\ 2 /
--------------------------------------
2
/ 1 \ / pi\
|-2 + -----------| *sec(x)*sec|x - --|
| / pi\| \ 2 /
| sec|x - --||
\ \ 2 //
$$\frac{4 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}{\left(-2 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ 4/x\ / 1 \ 3/x\ / 1 \ /x\
4*cos |-|*sin |-|*|-1 + -------| 4*sin |-|*|-1 + -------|*cos|-|
\2/ \2/ | 2/x\| \2/ | 2/x\| \2/
| tan |-|| | tan |-||
\ \2// \ \2//
-------------------------------- + -------------------------------
2 2 - sin(x)
(-2 + sin(x))
$$\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \sin^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
/ 2 \
2 | 2/x\ sin (x) |
sin (x)*|- sin |-| + ---------|
| \2/ 2/x\|
| 4*sin |-||
sin(2*x) \ \2//
- ----------- + -------------------------------
-2 + sin(x) 2
(-2 + sin(x))
$$\frac{\left(- \sin^{2}{\left(\frac{x}{2} \right)} + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} - \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)} - 2}$$
4 6
/ 2/x\\ 8/x\ / 2/x\\ /x\ / 2/x\\ 12/x\ 2/x\ / 2/x\\
4*|1 - tan |-|| *cos |-|*|1 - tan |-||*tan|-| 4*|1 - tan |-|| *cos |-|*tan |-|*|1 - tan |-||
\ \4// \4/ \ \2// \2/ \ \4// \4/ \2/ \ \2//
--------------------------------------------- + -----------------------------------------------
2 - sin(x) 2
(-2 + sin(x))
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{12}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
2/ pi\ / pi\
cos |x - --|*cos(x) 2*cos(x)*cos|x - --|
\ 2 / \ 2 /
------------------- + --------------------
2 / pi\
/ / pi\\ 2 - cos|x - --|
|2 - cos|x - --|| \ 2 /
\ \ 2 //
$$\frac{2 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{2 - \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(2 - \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}}$$
4 6
/ 2/x\\ 8/x\ / 2/x\\ /x\ / 2/x\\ 12/x\ 2/x\ / 2/x\\
4*|-1 + cot |-|| *sin |-|*|1 - tan |-||*tan|-| 4*|-1 + cot |-|| *sin |-|*tan |-|*|1 - tan |-||
\ \4// \4/ \ \2// \2/ \ \4// \4/ \2/ \ \2//
---------------------------------------------- + ------------------------------------------------
2 - sin(x) 2
(-2 + sin(x))
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
/ / pi\\ / pi\
|4 - cos|x - --||*cos(x)*cos|x - --|
\ \ 2 // \ 2 /
------------------------------------
2
/ / pi\\
|-2 + cos|x - --||
\ \ 2 //
$$\frac{\left(4 - \cos{\left(x - \frac{\pi}{2} \right)}\right) \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} - 2\right)^{2}}$$
3/x\ /x\
8*cos |-|*cos(x)*sin|-|
-cos(3*x) + cos(x) \2/ \2/
----------------------------- + -------------------------
2 (1 + cos(x))*(2 - sin(x))
2*(1 + cos(x))*(-2 + sin(x))
$$\frac{\cos{\left(x \right)} - \cos{\left(3 x \right)}}{2 \left(\sin{\left(x \right)} - 2\right)^{2} \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$
2/x\ / /x\ /x\ \
2*sin |-|*|-2 - cos|-|*sin|-| + sin(x)|*sin(2*x)
\2/ \ \2/ \2/ /
------------------------------------------------
2
(-1 + cos(x))*(-2 + sin(x))
$$\frac{2 \left(- \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} + \sin{\left(x \right)} - 2\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(2 x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2} \left(\cos{\left(x \right)} - 1\right)}$$
1 2
---------------------------- + --------------------------
2 / 1 \
/ 1 \ 2 |2 - ------|*csc(x)*sec(x)
|2 - ------| *csc (x)*sec(x) \ csc(x)/
\ csc(x)/
$$\frac{2}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \sec{\left(x \right)}} + \frac{1}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc^{2}{\left(x \right)} \sec{\left(x \right)}}$$
2/x\ / 2/x\\ / 2/x\\ /x\
4*tan |-|*|1 - tan |-|| 4*|1 - tan |-||*tan|-|
\2/ \ \2// \ \2// \2/
--------------------------------- + --------------------------------
2 / /x\ \
/ /x\ \ 2 | 2*tan|-| |
3 | 2*tan|-| | / 2/x\\ | \2/ |
/ 2/x\\ | \2/ | |1 + tan |-|| *|2 - -----------|
|1 + tan |-|| *|2 - -----------| \ \2// | 2/x\|
\ \2// | 2/x\| | 1 + tan |-||
| 1 + tan |-|| \ \2//
\ \2//
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
/ 2/x\ \ / 2/x\ \
| cos |-| | | cos |-| |
2/x\ 4/x pi\ | \2/ | 3/x pi\ | \2/ | /x\
4*cos |-|*cos |- - --|*|-1 + ------------| 4*cos |- - --|*|-1 + ------------|*cos|-|
\2/ \2 2 / | 2/x pi\| \2 2 / | 2/x pi\| \2/
| cos |- - --|| | cos |- - --||
\ \2 2 // \ \2 2 //
------------------------------------------ + -----------------------------------------
2 / pi\
/ / pi\\ 2 - cos|x - --|
|-2 + cos|x - --|| \ 2 /
\ \ 2 //
$$\frac{4 \left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} - 2\right)^{2}} + \frac{4 \left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 - \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
4*|1 - ------------| 4*|1 - ------------|
| 2/x pi\| | 2/x pi\|
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
---------------------------------------- + -------------------------------------
2 / 1 \ 3/x\ /x pi\
/ 1 \ 4/x\ 2/x pi\ |2 - -----------|*sec |-|*sec|- - --|
|-2 + -----------| *sec |-|*sec |- - --| | / pi\| \2/ \2 2 /
| / pi\| \2/ \2 2 / | sec|x - --||
| sec|x - --|| \ \ 2 //
\ \ 2 //
$$\frac{4 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(2 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{4 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(-2 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec^{4}{\left(\frac{x}{2} \right)} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
1
4 - ------
csc(x)
---------------------------------
2
/ 1 \ /pi \
|-2 + ------| *csc(x)*csc|-- - x|
\ csc(x)/ \2 /
$$\frac{4 - \frac{1}{\csc{\left(x \right)}}}{\left(-2 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
2/x\ 4/pi x\ | \2/| 4/x\ 6/pi x\ | \2/|
8*sin |-|*sin |-- + -|*|1 - ---------| 16*sin |-|*sin |-- + -|*|1 - ---------|
\2/ \2 2/ | 2 | \2/ \2 2/ | 2 |
\ sin (x) / \ sin (x) /
-------------------------------------- + ---------------------------------------
(2 - sin(x))*sin(x) 2 2
(-2 + sin(x)) *sin (x)
$$\frac{16 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2} \sin^{2}{\left(x \right)}} + \frac{8 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(2 - \sin{\left(x \right)}\right) \sin{\left(x \right)}}$$
1 2
-------------------------------------- + ------------------------------------
2 / 1 \ / pi\
/ 1 \ 2/ pi\ |2 - -----------|*sec(x)*sec|x - --|
|2 - -----------| *sec(x)*sec |x - --| | / pi\| \ 2 /
| / pi\| \ 2 / | sec|x - --||
| sec|x - --|| \ \ 2 //
\ \ 2 //
$$\frac{2}{\left(2 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(2 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$
2 / pi\ / pi\
sin (x)*sin|x + --| 2*sin(x)*sin|x + --|
\ 2 / \ 2 /
------------------- + --------------------
2 2 - sin(x)
(2 - sin(x))
$$\frac{2 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{2 - \sin{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
4*|1 - ------------| 4*|1 - ------------|
| 2/x\ | | 2/x\ |
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
----------------------------------- + --------------------------------
2 / 1 \ /x\ 3/pi x\
/ 1 \ 2/x\ 4/pi x\ |2 - ------|*csc|-|*csc |-- - -|
|-2 + ------| *csc |-|*csc |-- - -| \ csc(x)/ \2/ \2 2/
\ csc(x)/ \2/ \2 2/
$$\frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{4 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(-2 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} \right)} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
1 2
--------------------------------- + -------------------------------
2 / 1 \ /pi \
/ 1 \ 2 /pi \ |2 - ------|*csc(x)*csc|-- - x|
|2 - ------| *csc (x)*csc|-- - x| \ csc(x)/ \2 /
\ csc(x)/ \2 /
$$\frac{2}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc^{2}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 2 \ / 2 \
2 2/x\ | sin (x) | 2/x\ | sin (x) |
sin (x)*sin |-|*|-1 + ---------| 2*sin |-|*|-1 + ---------|*sin(x)
\2/ | 4/x\| \2/ | 4/x\|
| 4*sin |-|| | 4*sin |-||
\ \2// \ \2//
-------------------------------- + ---------------------------------
2 2 - sin(x)
(-2 + sin(x))
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{2 \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{2 - \sin{\left(x \right)}}$$
/ /x\ \
| 2*cot|-| |
/ 2/x\\ | \2/ | /x\
2*|-1 + cot |-||*|4 - -----------|*cot|-|
\ \2// | 2/x\| \2/
| 1 + cot |-||
\ \2//
-----------------------------------------
2
/ /x\ \
2 | 2*cot|-| |
/ 2/x\\ | \2/ |
|1 + cot |-|| *|-2 + -----------|
\ \2// | 2/x\|
| 1 + cot |-||
\ \2//
$$\frac{2 \left(4 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(-2 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
6/x\ 2/x\ / 2/x\\ 4/x\ / 2/x\\ /x\
4*cos |-|*tan |-|*|1 - tan |-|| 4*cos |-|*|1 - tan |-||*tan|-|
\2/ \2/ \ \2// \2/ \ \2// \2/
------------------------------- + ------------------------------
2 2 - sin(x)
(-2 + sin(x))
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{6}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
4/x\ 8/x\ / 2/x\\ /x\ 2/x\ 6/x\ 12/x\ / 2/x\\
64*cot |-|*sin |-|*|-1 + cot |-||*cot|-| 256*cot |-|*cot |-|*sin |-|*|-1 + cot |-||
\4/ \4/ \ \2// \2/ \2/ \4/ \4/ \ \2//
---------------------------------------- + -------------------------------------------
2 - sin(x) 2
(-2 + sin(x))
$$\frac{256 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{64 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{8}{\left(\frac{x}{4} \right)} \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
4/x\ / 2/x\\ /x\ 2/x\ 6/x\ / 2/x\\
64*cot |-|*|-1 + cot |-||*cot|-| 256*cot |-|*cot |-|*|-1 + cot |-||
\4/ \ \2// \2/ \2/ \4/ \ \2//
-------------------------------- + ----------------------------------
/ /x\ \ 2
4 | 2*cot|-| | / /x\ \
/ 2/x\\ | \2/ | 6 | 2*cot|-| |
|1 + cot |-|| *|2 - -----------| / 2/x\\ | \2/ |
\ \4// | 2/x\| |1 + cot |-|| *|-2 + -----------|
| 1 + cot |-|| \ \4// | 2/x\|
\ \2// | 1 + cot |-||
\ \2//
$$\frac{64 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{256 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot^{6}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(-2 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}}$$
/ /x\ \
| 2*tan|-| |
/ 2/x\\ | \2/ | /x\
2*|1 - tan |-||*|4 - -----------|*tan|-|
\ \2// | 2/x\| \2/
| 1 + tan |-||
\ \2//
----------------------------------------
2
/ /x\ \
2 | 2*tan|-| |
/ 2/x\\ | \2/ |
|1 + tan |-|| *|-2 + -----------|
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(4 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \tan{\left(\frac{x}{2} \right)}}{\left(-2 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
6 4
/ 2/x\\ 2/x\ / 2/x\\ / 2/x\\ / 2/x\\ /x\
4*|1 - tan |-|| *tan |-|*|1 - tan |-|| 4*|1 - tan |-|| *|1 - tan |-||*tan|-|
\ \4// \2/ \ \2// \ \4// \ \2// \2/
-------------------------------------- + -------------------------------------
2 / /x\ \
/ /x\ \ 4 | 2*tan|-| |
6 | 2*tan|-| | / 2/x\\ | \2/ |
/ 2/x\\ | \2/ | |1 + tan |-|| *|2 - -----------|
|1 + tan |-|| *|-2 + -----------| \ \4// | 2/x\|
\ \4// | 2/x\| | 1 + tan |-||
| 1 + tan |-|| \ \2//
\ \2//
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\left(-2 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}}$$
4/x\ / 2/x\\ /x\
2*cos |-|*|1 - tan |-||*(4 - sin(x))*tan|-|
\2/ \ \2// \2/
-------------------------------------------
2
(-2 + sin(x))
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(4 - \sin{\left(x \right)}\right) \cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
/ pi\
(4 - sin(x))*sin(x)*sin|x + --|
\ 2 /
-------------------------------
2
(-2 + sin(x))
$$\frac{\left(4 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
2/x\ / 2/x\\ / 2/x\\ /x\
4*cot |-|*|-1 + cot |-|| 4*|-1 + cot |-||*cot|-|
\2/ \ \2// \ \2// \2/
--------------------------------- + --------------------------------
2 / /x\ \
/ /x\ \ 2 | 2*cot|-| |
3 | 2*cot|-| | / 2/x\\ | \2/ |
/ 2/x\\ | \2/ | |1 + cot |-|| *|2 - -----------|
|1 + cot |-|| *|2 - -----------| \ \2// | 2/x\|
\ \2// | 2/x\| | 1 + cot |-||
| 1 + cot |-|| \ \2//
\ \2//
$$\frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot^{2}{\left(\frac{x}{2} \right)}}{\left(2 - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
/ 2/x pi\\ / 2/x pi\\
| sec |- - --|| | sec |- - --||
| \2 2 /| | \2 2 /|
4*|-1 + ------------| 4*|-1 + ------------|
| 2/x\ | | 2/x\ |
| sec |-| | | sec |-| |
\ \2/ / \ \2/ /
---------------------------------------- + -------------------------------------
2 / 1 \ /x\ 3/x pi\
/ 1 \ 2/x\ 4/x pi\ |2 - -----------|*sec|-|*sec |- - --|
|-2 + -----------| *sec |-|*sec |- - --| | / pi\| \2/ \2 2 /
| / pi\| \2/ \2 2 / | sec|x - --||
| sec|x - --|| \ \ 2 //
\ \ 2 //
$$\frac{4 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(2 - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{4 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(-2 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
cos(x) cos(3*x)
2*sin(2*x) - ------ + --------
4 4
------------------------------
2
(-2 + sin(x))
$$\frac{2 \sin{\left(2 x \right)} - \frac{\cos{\left(x \right)}}{4} + \frac{\cos{\left(3 x \right)}}{4}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
2
sin (x)*cos(x) 2*cos(x)*sin(x)
-------------- + ---------------
2 2 - sin(x)
(2 - sin(x))
$$\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{2 - \sin{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right)^{2}}$$
/ 2/x\ \ / 2/x\ \
| csc |-| | | csc |-| |
| \2/ | | \2/ |
4*|-1 + ------------| 4*|-1 + ------------|
| 2/pi x\| | 2/pi x\|
| csc |-- - -|| | csc |-- - -||
\ \2 2// \ \2 2//
----------------------------------- + --------------------------------
2 / 1 \ 3/x\ /pi x\
/ 1 \ 4/x\ 2/pi x\ |2 - ------|*csc |-|*csc|-- - -|
|-2 + ------| *csc |-|*csc |-- - -| \ csc(x)/ \2/ \2 2/
\ csc(x)/ \2/ \2 2/
$$\frac{4 \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)}{\left(2 - \frac{1}{\csc{\left(x \right)}}\right) \csc^{3}{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{4 \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)}{\left(-2 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc^{4}{\left(\frac{x}{2} \right)} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
4/x\ 2/x\ / 2/x\\ 3/x\ / 2/x\\ /x\
4*cos |-|*sin |-|*|1 - tan |-|| 4*cos |-|*|1 - tan |-||*sin|-|
\2/ \2/ \ \2// \2/ \ \2// \2/
------------------------------- + ------------------------------
2 2 - sin(x)
(-2 + sin(x))
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
4/x\ 2/x pi\ | \2 2 /| 3/x\ | \2 2 /| /x pi\
4*cos |-|*cos |- - --|*|1 - ------------| 4*cos |-|*|1 - ------------|*cos|- - --|
\2/ \2 2 / | 2/x\ | \2/ | 2/x\ | \2 2 /
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
----------------------------------------- + ----------------------------------------
2 / pi\
/ / pi\\ 2 - cos|x - --|
|-2 + cos|x - --|| \ 2 /
\ \ 2 //
$$\frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} - 2\right)^{2}} + \frac{4 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 - \cos{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ 6/x\ / 2/x\\ 4/x\ / 2/x\\ /x\
4*cot |-|*sin |-|*|-1 + cot |-|| 4*sin |-|*|-1 + cot |-||*cot|-|
\2/ \2/ \ \2// \2/ \ \2// \2/
-------------------------------- + -------------------------------
2 2 - sin(x)
(-2 + sin(x))
$$\frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{6}{\left(\frac{x}{2} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2 - \sin{\left(x \right)}}$$
4/x\ / 2/x\\ /x\
2*sin |-|*|-1 + cot |-||*(4 - sin(x))*cot|-|
\2/ \ \2// \2/
--------------------------------------------
2
(-2 + sin(x))
$$\frac{2 \left(4 - \sin{\left(x \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
4/x\ / 1 \ 6/x\ / 1 \
64*tan |-|*|-1 + -------| 256*tan |-|*|-1 + -------|
\4/ | 2/x\| \4/ | 2/x\|
| tan |-|| | tan |-||
\ \2// \ \2//
--------------------------------------- + ------------------------------------------
/ /x\ \ 2
4 | 2*tan|-| | / /x\ \
/ 2/x\\ | \2/ | /x\ 6 | 2*tan|-| |
|1 + tan |-|| *|2 - -----------|*tan|-| / 2/x\\ | \2/ | 2/x\
\ \4// | 2/x\| \2/ |1 + tan |-|| *|-2 + -----------| *tan |-|
| 1 + tan |-|| \ \4// | 2/x\| \2/
\ \2// | 1 + tan |-||
\ \2//
$$\frac{64 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan^{4}{\left(\frac{x}{4} \right)}}{\left(2 - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} \right)}} + \frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan^{6}{\left(\frac{x}{4} \right)}}{\left(-2 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \tan^{2}{\left(\frac{x}{2} \right)}}$$
(4 - sin(x))*cos(x)*sin(x)
--------------------------
2
(-2 + sin(x))
$$\frac{\left(4 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2}}$$
8/x\ 4/x\ / 1 \ 12/x\ 6/x\ / 1 \
64*cos |-|*tan |-|*|-1 + -------| 256*cos |-|*tan |-|*|-1 + -------|
\4/ \4/ | 2/x\| \4/ \4/ | 2/x\|
| tan |-|| | tan |-||
\ \2// \ \2//
--------------------------------- + -----------------------------------
/x\ 2 2/x\
(2 - sin(x))*tan|-| (-2 + sin(x)) *tan |-|
\2/ \2/
$$\frac{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{12}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\sin{\left(x \right)} - 2\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{64 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(2 - \sin{\left(x \right)}\right) \tan{\left(\frac{x}{2} \right)}}$$
64*cos(x/4)^8*tan(x/4)^4*(-1 + tan(x/2)^(-2))/((2 - sin(x))*tan(x/2)) + 256*cos(x/4)^12*tan(x/4)^6*(-1 + tan(x/2)^(-2))/((-2 + sin(x))^2*tan(x/2)^2)