Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sinx/(((1+tgx/2)/(1-tgx/2))*(1+sinx))
¿Cómo vas a descomponer esta sinx/(((1+tgx/2)/(1-tgx/2))*(1+sinx)) expresión en fracciones?
Solución
Ha introducido
[src]
sin(x)
-----------------------
tan(x)
1 + ------
2
----------*(1 + sin(x))
tan(x)
1 - ------
2
$$\frac{\sin{\left(x \right)}}{\frac{\frac{\tan{\left(x \right)}}{2} + 1}{- \frac{\tan{\left(x \right)}}{2} + 1} \left(\sin{\left(x \right)} + 1\right)}$$
sin(x)/((((1 + tan(x)/2)/(1 - tan(x)/2))*(1 + sin(x))))
Simplificación general
[src]
-(-2 + tan(x))*sin(x) ------------------------- (1 + sin(x))*(2 + tan(x))
$$- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
-(-2 + tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))
Respuesta numérica
[src]
(1.0 - 0.5*tan(x))*sin(x)/((1.0 + 0.5*tan(x))*(1.0 + sin(x)))
(1.0 - 0.5*tan(x))*sin(x)/((1.0 + 0.5*tan(x))*(1.0 + sin(x)))
Potencias
[src]
/ / I*x -I*x\\
| I*\- e + e /| / -I*x I*x\
-I*|1 - ------------------|*\- e + e /
| / I*x -I*x\ |
\ 2*\e + e / /
---------------------------------------------------
/ / -I*x I*x\\ / / I*x -I*x\\
| I*\- e + e /| | I*\- e + e /|
2*|1 - ------------------|*|1 + ------------------|
\ 2 / | / I*x -I*x\ |
\ 2*\e + e / /
$$- \frac{i \left(- \frac{i \left(- e^{i x} + e^{- i x}\right)}{2 \left(e^{i x} + e^{- i x}\right)} + 1\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right) \left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{2 \left(e^{i x} + e^{- i x}\right)} + 1\right)}$$
/ tan(x)\ |1 - ------|*sin(x) \ 2 / ------------------------- / tan(x)\ |1 + ------|*(1 + sin(x)) \ 2 /
$$\frac{\left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\frac{\tan{\left(x \right)}}{2} + 1\right)}$$
(1 - tan(x)/2)*sin(x)/((1 + tan(x)/2)*(1 + sin(x)))
Combinatoria
[src]
-(-2 + tan(x))*sin(x) ------------------------- (1 + sin(x))*(2 + tan(x))
$$- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
-(-2 + tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))
Denominador común
[src]
2 + 4*sin(x) + tan(x)
-1 + -------------------------------------
2 + 2*sin(x) + sin(x)*tan(x) + tan(x)
$$\frac{4 \sin{\left(x \right)} + \tan{\left(x \right)} + 2}{\sin{\left(x \right)} \tan{\left(x \right)} + 2 \sin{\left(x \right)} + \tan{\left(x \right)} + 2} - 1$$
-1 + (2 + 4*sin(x) + tan(x))/(2 + 2*sin(x) + sin(x)*tan(x) + tan(x))
Denominador racional
[src]
(4 - 2*tan(x))*sin(x) --------------------------- (1 + sin(x))*(4 + 2*tan(x))
$$\frac{\left(4 - 2 \tan{\left(x \right)}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(2 \tan{\left(x \right)} + 4\right)}$$
(4 - 2*tan(x))*sin(x)/((1 + sin(x))*(4 + 2*tan(x)))
Unión de expresiones racionales
[src]
(2 - tan(x))*sin(x) ------------------------- (1 + sin(x))*(2 + tan(x))
$$\frac{\left(2 - \tan{\left(x \right)}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
(2 - tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))
Compilar la expresión
[src]
/ tan(x)\ |1 - ------|*sin(x) \ 2 / ------------------------- / tan(x)\ |1 + ------|*(1 + sin(x)) \ 2 /
$$\frac{\left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\frac{\tan{\left(x \right)}}{2} + 1\right)}$$
(1 - tan(x)/2)*sin(x)/((1 + tan(x)/2)*(1 + sin(x)))
Parte trigonométrica
[src]
/ tan(x)\ /x\
4*|1 - ------|*tan|-|
\ 2 / \2/
--------------------------------------------
/ /x\ \
| 2*tan|-| |
/ 2/x\\ | \2/ |
|1 + tan |-||*|1 + -----------|*(2 + tan(x))
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ sin(x) \ |1 - --------|*sin(x) \ 2*cos(x)/ --------------------------- / sin(x) \ |1 + --------|*(1 + sin(x)) \ 2*cos(x)/
$$\frac{\left(- \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)}$$
/ tan(x)\
2*|1 - ------|*sin(x)
\ 2 /
-------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
2/x\ 2/x\ / tan(x)\
8*cos |-|*sin |-|*|1 - ------|
\2/ \2/ \ 2 /
--------------------------------
(1 + sin(x))*(2 + tan(x))*sin(x)
$$\frac{8 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right) \sin{\left(x \right)}}$$
2
/ 2/x\\ / tan(x)\ /x\
4*|1 - tan |-|| *|1 - ------|*tan|-|
\ \4// \ 2 / \2/
---------------------------------------------
/ /x\ \
2 | 2*tan|-| |
/ 2/x\\ | \2/ |
|1 + tan |-|| *|1 + -----------|*(2 + tan(x))
\ \4// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan{\left(x \right)} + 2\right)}$$
4/x\ 2/x\ / tan(x)\
16*cos |-|*tan |-|*|1 - ------|
\4/ \4/ \ 2 /
--------------------------------
/x\
(1 + sin(x))*(2 + tan(x))*tan|-|
\2/
$$\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}$$
/ tan(x)\ /x\
2*|1 - ------|*tan|-|
\ 2 / \2/
--------------------------------------------
/ /x\ \
| 2*tan|-| |
/ 2/x\\ / tan(x)\ | \2/ |
|1 + tan |-||*|1 + ------|*|1 + -----------|
\ \2// \ 2 / | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\frac{\tan{\left(x \right)}}{2} + 1\right)}$$
/ 2 \
2/x\ 2/pi x\ | sin (x) |
8*sin |-|*sin |-- + -|*|1 - --------|
\2/ \2 2/ \ sin(2*x)/
-------------------------------------
/ 2 \
| 2*sin (x)|
(1 + sin(x))*|2 + ---------|*sin(x)
\ sin(2*x)/
$$\frac{8 \left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 2\right) \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}$$
/ 1 \ /x\
2*|1 - --------|*cot|-|
\ 2*cot(x)/ \2/
----------------------------------------------
/ /x\ \
| 2*cot|-| |
/ 2/x\\ / 1 \ | \2/ |
|1 + cot |-||*|1 + --------|*|1 + -----------|
\ \2// \ 2*cot(x)/ | 2/x\|
| 1 + cot |-||
\ \2//
$$\frac{2 \left(1 - \frac{1}{2 \cot{\left(x \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 + \frac{1}{2 \cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2 \ | sin (x) | |1 - --------|*sin(x) \ sin(2*x)/ --------------------------- / 2 \ | sin (x) | |1 + --------|*(1 + sin(x)) \ sin(2*x)/
$$\frac{\left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)}$$
/pi \
csc|-- - x|
\2 /
1 - -----------
2*csc(x)
-------------------------------------
/ /pi \\
| csc|-- - x||
/ 1 \ | \2 /|
|1 + ------|*|1 + -----------|*csc(x)
\ csc(x)/ \ 2*csc(x) /
$$\frac{1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
sec(x)
1 - --------
2*csc(x)
----------------------------------
/ 1 \ / sec(x) \
|1 + ------|*|1 + --------|*csc(x)
\ csc(x)/ \ 2*csc(x)/
$$\frac{1 - \frac{\sec{\left(x \right)}}{2 \csc{\left(x \right)}}}{\left(1 + \frac{\sec{\left(x \right)}}{2 \csc{\left(x \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
2
/ 2/x\\ 4/x\ / tan(x)\ /x\
4*|1 - tan |-|| *cos |-|*|1 - ------|*tan|-|
\ \4// \4/ \ 2 / \2/
--------------------------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ sec(x) \
2*|1 - -------------|
| / pi\|
| 2*sec|x - --||
\ \ 2 //
-----------------------------------------------
/ 1 \ / sec(x) \ / pi\
|1 + -----------|*|2 + -----------|*sec|x - --|
| / pi\| | / pi\| \ 2 /
| sec|x - --|| | sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{2 \left(- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
sec(x)
1 - -------------
/ pi\
2*sec|x - --|
\ 2 /
-------------------------------------------------
/ 1 \ / sec(x) \ / pi\
|1 + -----------|*|1 + -------------|*sec|x - --|
| / pi\| | / pi\| \ 2 /
| sec|x - --|| | 2*sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ / tan(x)\
16*tan |-|*|1 - ------|
\4/ \ 2 /
----------------------------------------------------
/ /x\ \
2 | 2*tan|-| |
/ 2/x\\ | \2/ | /x\
|1 + tan |-|| *|1 + -----------|*(2 + tan(x))*tan|-|
\ \4// | 2/x\| \2/
| 1 + tan |-||
\ \2//
$$\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan{\left(x \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}$$
2/x\ / 1 \ /x\
16*cot |-|*|1 - --------|*cot|-|
\4/ \ 2*cot(x)/ \2/
---------------------------------------------
/ /x\ \
2 | 2*cot|-| |
/ 2/x\\ | \2/ | / 1 \
|1 + cot |-|| *|1 + -----------|*|2 + ------|
\ \4// | 2/x\| \ cot(x)/
| 1 + cot |-||
\ \2//
$$\frac{16 \left(1 - \frac{1}{2 \cot{\left(x \right)}}\right) \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
2
/ 2/x\\ / 1 \
4*|-1 + cot |-|| *|1 - --------|
\ \4// \ 2*cot(x)/
----------------------------------------------------
/ /x\ \
2 | 2*cot|-| |
/ 2/x\\ | \2/ | / 1 \ /x\
|1 + cot |-|| *|1 + -----------|*|2 + ------|*cot|-|
\ \4// | 2/x\| \ cot(x)/ \2/
| 1 + cot |-||
\ \2//
$$\frac{4 \left(1 - \frac{1}{2 \cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\\ 4/x\ / tan(x)\ /x\
4*|-1 + cot |-|| *sin |-|*|1 - ------|*tan|-|
\ \4// \4/ \ 2 / \2/
---------------------------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ sec(x) \
4*|1 - -------------|
| / pi\|
| 2*sec|x - --||
\ \ 2 //
------------------------------------------------------
/ 1 \ / sec(x) \ /x\ /x pi\
|1 + -----------|*|2 + -----------|*sec|-|*sec|- - --|
| / pi\| | / pi\| \2/ \2 2 /
| sec|x - --|| | sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{4 \left(- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ / pi\\ | cos|x - --|| | \ 2 /| / pi\ |1 - -----------|*cos|x - --| \ 2*cos(x) / \ 2 / ----------------------------------- / / pi\\ | cos|x - --|| | \ 2 /| / / pi\\ |1 + -----------|*|1 + cos|x - --|| \ 2*cos(x) / \ \ 2 //
$$\frac{\left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}$$
/ 1 \ /x\
4*|1 - --------|*cot|-|
\ 2*cot(x)/ \2/
--------------------------------------------
/ /x\ \
| 2*cot|-| |
/ 2/x\\ | \2/ | / 1 \
|1 + cot |-||*|1 + -----------|*|2 + ------|
\ \2// | 2/x\| \ cot(x)/
| 1 + cot |-||
\ \2//
$$\frac{4 \left(1 - \frac{1}{2 \cot{\left(x \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ /pi \\
| csc|-- - x||
| \2 /|
4*|1 - -----------|
\ 2*csc(x) /
-------------------------------------------------
/ /pi \\
| csc|-- - x||
/ 1 \ | \2 /| /x\ /pi x\
|1 + ------|*|2 + -----------|*csc|-|*csc|-- - -|
\ csc(x)/ \ csc(x) / \2/ \2 2/
$$\frac{4 \left(1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right)}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(2 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
-(-2 + tan(x))*sin(x) ------------------------- (1 + sin(x))*(2 + tan(x))
$$- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ / pi\\
| cos|x - --||
| \ 2 /| / pi\
2*|1 - -----------|*cos|x - --|
\ 2*cos(x) / \ 2 /
-----------------------------------
/ / pi\\
| cos|x - --||
/ / pi\\ | \ 2 /|
|1 + cos|x - --||*|2 + -----------|
\ \ 2 // \ cos(x) /
$$\frac{2 \left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(2 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}$$
2/x\ / tan(x)\ /x\
4*cos |-|*|1 - ------|*tan|-|
\2/ \ 2 / \2/
-----------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ /pi \\
| csc|-- - x||
| \2 /|
2*|1 - -----------|
\ 2*csc(x) /
-------------------------------------
/ /pi \\
| csc|-- - x||
/ 1 \ | \2 /|
|1 + ------|*|2 + -----------|*csc(x)
\ csc(x)/ \ csc(x) /
$$\frac{2 \left(1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right)}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(2 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/ 2 \
| sin (x) |
2*|1 - --------|*sin(x)
\ sin(2*x)/
----------------------------
/ 2 \
| 2*sin (x)|
(1 + sin(x))*|2 + ---------|
\ sin(2*x)/
$$\frac{2 \left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 2\right) \left(\sin{\left(x \right)} + 1\right)}$$
2/x\ 4/x\ / tan(x)\ /x\
16*cot |-|*sin |-|*|1 - ------|*cot|-|
\4/ \4/ \ 2 / \2/
--------------------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{4}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ / pi\\
| cos|x - --||
| \ 2 /| /x\ /x pi\
4*|1 - -----------|*cos|-|*cos|- - --|
\ 2*cos(x) / \2/ \2 2 /
--------------------------------------
/ / pi\\
| cos|x - --||
/ / pi\\ | \ 2 /|
|1 + cos|x - --||*|2 + -----------|
\ \ 2 // \ cos(x) /
$$\frac{4 \left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(2 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}$$
2/x\ / tan(x)\ /x\
4*sin |-|*|1 - ------|*cot|-|
\2/ \ 2 / \2/
-----------------------------
(1 + sin(x))*(2 + tan(x))
$$\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
/ tan(x)\ /x\ /x\ 4*|1 - ------|*cos|-|*sin|-| \ 2 / \2/ \2/ ---------------------------- (1 + sin(x))*(2 + tan(x))
$$\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}$$
4*(1 - tan(x)/2)*cos(x/2)*sin(x/2)/((1 + sin(x))*(2 + tan(x)))