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