Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)*(-sin(x)-(cos(x)*sin(x)))/sqrt(1+cos(x)^(2))/(cos(x)+sqrt(1+cos(x)^(2)))
¿Cómo vas a descomponer esta cos(x)*(-sin(x)-(cos(x)*sin(x)))/sqrt(1+cos(x)^(2))/(cos(x)+sqrt(1+cos(x)^(2))) expresión en fracciones?
Solución
Ha introducido
[src]
/cos(x)*(-sin(x) - cos(x)*sin(x))\
|--------------------------------|
| _____________ |
| / 2 |
\ \/ 1 + cos (x) /
----------------------------------
_____________
/ 2
cos(x) + \/ 1 + cos (x)
$$\frac{\left(- \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}\right) \cos{\left(x \right)} \frac{1}{\sqrt{\cos^{2}{\left(x \right)} + 1}}}{\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}}$$
((cos(x)*(-sin(x) - cos(x)*sin(x)))/sqrt(1 + cos(x)^2))/(cos(x) + sqrt(1 + cos(x)^2))
Simplificación general
[src]
-(1 + cos(x))*cos(x)*sin(x)
-------------------------------------
_____________
2 / 2
1 + cos (x) + \/ 1 + cos (x) *cos(x)
$$- \frac{\left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} + 1}$$
-(1 + cos(x))*cos(x)*sin(x)/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x))
Respuesta numérica
[src]
(1.0 + cos(x)^2)^(-0.5)*(-sin(x) - cos(x)*sin(x))*cos(x)/((1.0 + cos(x)^2)^0.5 + cos(x))
(1.0 + cos(x)^2)^(-0.5)*(-sin(x) - cos(x)*sin(x))*cos(x)/((1.0 + cos(x)^2)^0.5 + cos(x))
Combinatoria
[src]
-(1 + cos(x))*cos(x)*sin(x) -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
-(1 + cos(x))*cos(x)*sin(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))
Denominador común
[src]
_____________
/ 2
-cos(x)*sin(x) + \/ 1 + cos (x) *cos(x)*sin(x) + sin(x)
-sin(x) + --------------------------------------------------------
_____________
2 / 2
1 + cos (x) + \/ 1 + cos (x) *cos(x)
$$- \sin{\left(x \right)} + \frac{\sqrt{\cos^{2}{\left(x \right)} + 1} \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} + 1}$$
-sin(x) + (-cos(x)*sin(x) + sqrt(1 + cos(x)^2)*cos(x)*sin(x) + sin(x))/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x))
Denominador racional
[src]
_____________ _____________
2 3 / 2 2 / 2
cos (x)*sin(x) + cos (x)*sin(x) - \/ 1 + cos (x) *cos (x)*sin(x) - \/ 1 + cos (x) *cos(x)*sin(x)
--------------------------------------------------------------------------------------------------
_____________
/ 2
\/ 1 + cos (x)
$$\frac{- \sqrt{\cos^{2}{\left(x \right)} + 1} \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \sqrt{\cos^{2}{\left(x \right)} + 1} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos^{3}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1}}$$
(cos(x)^2*sin(x) + cos(x)^3*sin(x) - sqrt(1 + cos(x)^2)*cos(x)^2*sin(x) - sqrt(1 + cos(x)^2)*cos(x)*sin(x))/sqrt(1 + cos(x)^2)
Potencias
[src]
(-sin(x) - cos(x)*sin(x))*cos(x) -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$\frac{\left(- \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ / I*x -I*x\ \
| |e e | / -I*x I*x\|
/ I*x -I*x\ | / -I*x I*x\ I*|---- + -----|*\- e + e /|
|e e | |I*\- e + e / \ 2 2 / |
|---- + -----|*|------------------ + ---------------------------------|
\ 2 2 / \ 2 2 /
------------------------------------------------------------------------
_____________________ / _____________________ \
/ 2 | / 2 |
/ / I*x -I*x\ | / / I*x -I*x\ I*x -I*x|
/ |e e | | / |e e | e e |
/ 1 + |---- + -----| *| / 1 + |---- + -----| + ---- + -----|
\/ \ 2 2 / \\/ \ 2 2 / 2 2 /
$$\frac{\left(\frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2} + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{\sqrt{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + 1} \left(\sqrt{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} + 1} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}$$
(exp(i*x)/2 + exp(-i*x)/2)*(i*(-exp(-i*x) + exp(i*x))/2 + i*(exp(i*x)/2 + exp(-i*x)/2)*(-exp(-i*x) + exp(i*x))/2)/(sqrt(1 + (exp(i*x)/2 + exp(-i*x)/2)^2)*(sqrt(1 + (exp(i*x)/2 + exp(-i*x)/2)^2) + exp(i*x)/2 + exp(-i*x)/2))
Unión de expresiones racionales
[src]
(-1 - cos(x))*cos(x)*sin(x) -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$\frac{\left(- \cos{\left(x \right)} - 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
(-1 - cos(x))*cos(x)*sin(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))
Compilar la expresión
[src]
(-sin(x) - cos(x)*sin(x))*cos(x) -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$\frac{\left(- \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
(-sin(x) - cos(x)*sin(x))*cos(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))
Parte trigonométrica
[src]
/ 2/x\\
| -1 + cot |-||
2/x\ | \2/| / 2/x\\ /x\
-8*cot |-|*|1 + ------------|*|-1 + cot |-||*cot|-|
\4/ | 2/x\ | \ \2// \2/
| 1 + cot |-| |
\ \2/ /
-------------------------------------------------------------------------------------------------------------
_____________________ / _____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
2 / |-1 + cot |-|| | / |-1 + cot |-|| -1 + cot |-||
/ 2/x\\ / 2/x\\ / \ \2// | / \ \2// \2/|
|1 + cot |-||*|1 + cot |-|| * / 1 + --------------- *| / 1 + --------------- + ------------|
\ \2// \ \4// / 2 | / 2 2/x\ |
/ / 2/x\\ | / / 2/x\\ 1 + cot |-| |
/ |1 + cot |-|| | / |1 + cot |-|| \2/ |
\/ \ \2// \\/ \ \2// /
$$- \frac{8 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2
/ 2/x\\ 2/x\ 4/x\ / 2/x\\ /x\
-2*|1 - tan |-|| *cos |-|*cos |-|*|1 - tan |-||*(1 + cos(x))*tan|-|
\ \4// \2/ \4/ \ \2// \2/
-------------------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cos{\left(x \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ /x\ / 2/x\\ /x\\
| 2*cot|-| 2*|-1 + cot |-||*cot|-||
/ 2/x\\ | \2/ \ \2// \2/|
|-1 + cot |-||*|- ----------- - -----------------------|
\ \2// | 2/x\ 2 |
| 1 + cot |-| / 2/x\\ |
| \2/ |1 + cot |-|| |
\ \ \2// /
----------------------------------------------------------------------------------------------
_____________________ / _____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
/ |-1 + cot |-|| | / |-1 + cot |-|| -1 + cot |-||
/ 2/x\\ / \ \2// | / \ \2// \2/|
|1 + cot |-||* / 1 + --------------- *| / 1 + --------------- + ------------|
\ \2// / 2 | / 2 2/x\ |
/ / 2/x\\ | / / 2/x\\ 1 + cot |-| |
/ |1 + cot |-|| | / |1 + cot |-|| \2/ |
\/ \ \2// \\/ \ \2// /
$$\frac{\left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x pi\\
| cos |- - --||
2/x\ | \2 2 /| / pi\
-cos |-|*|1 - ------------|*(1 + cos(x))*cos|x - --|
\2/ | 2/x\ | \ 2 /
| cos |-| |
\ \2/ /
-----------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cos{\left(x \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ 2 \
2/x\ / / pi\\ | sin (x) |
-sin |-|*|1 + sin|x + --||*|-1 + ---------|*sin(x)
\2/ \ \ 2 // | 4/x\|
| 4*sin |-||
\ \2//
-------------------------------------------------------------
__________________ / __________________ \
/ 2/ pi\ | / 2/ pi\ / pi\|
/ 1 + sin |x + --| *| / 1 + sin |x + --| + sin|x + --||
\/ \ 2 / \\/ \ 2 / \ 2 //
$$- \frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{\left(\sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1} + \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1}}$$
2/x\ / 1 \
-sin |-|*(1 + cos(x))*|-1 + -------|*sin(x)
\2/ | 2/x\|
| tan |-||
\ \2//
--------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ 1 \
-|1 + ------|
\ sec(x)/
-----------------------------------------------------
/ _____________\
| / 1 |
| / 1 + ------- |
| / 2 |
| 1 \/ sec (x) | / pi\
|1 + ------- + ------------------|*sec(x)*sec|x - --|
| 2 sec(x) | \ 2 /
\ sec (x) /
$$- \frac{1 + \frac{1}{\sec{\left(x \right)}}}{\left(\frac{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}}}{\sec{\left(x \right)}} + 1 + \frac{1}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ 2/x pi\\
| sec |- - --||
/ 1 \ | \2 2 /|
-|1 + ------|*|-1 + ------------|
\ sec(x)/ | 2/x\ |
| sec |-| |
\ \2/ /
-------------------------------------------------------------------------
_____________ / _____________ \
/ 1 | / 1 1 | / pi\ 2/x pi\
/ 1 + ------- *| / 1 + ------- + ------|*sec|x - --|*sec |- - --|
/ 2 | / 2 sec(x)| \ 2 / \2 2 /
\/ sec (x) \\/ sec (x) /
$$- \frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\sec{\left(x \right)}}\right)}{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} \left(\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ 4/x\ 2/x\ / 1 \ /x\
-8*cos |-|*cos |-|*tan |-|*(1 + cos(x))*|-1 + -------|*tan|-|
\2/ \4/ \4/ | 2/x\| \2/
| tan |-||
\ \2//
-------------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{8 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cos{\left(x \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \cos^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ 1 \
-|1 + -----------|
| /pi \|
| csc|-- - x||
\ \2 //
----------------------------------------------------------------
/ __________________\
| / 1 |
| / 1 + ------------ |
| / 2/pi \ |
| / csc |-- - x| |
| 1 \/ \2 / | /pi \
|1 + ------------ + ------------------------|*csc(x)*csc|-- - x|
| 2/pi \ /pi \ | \2 /
| csc |-- - x| csc|-- - x| |
\ \2 / \2 / /
$$- \frac{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(\frac{\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + 1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 2/x\\
2 | -1 + cot |-||
/ 2/x\\ / 1 \ | \2/| /x\
-2*|-1 + cot |-|| *|1 - -------|*|1 + ------------|*cot|-|
\ \4// | 2/x\| | 2/x\ | \2/
| cot |-|| | 1 + cot |-| |
\ \2// \ \2/ /
-------------------------------------------------------------------------------------------------------------
_____________________ / _____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
2 / |-1 + cot |-|| | / |-1 + cot |-|| -1 + cot |-||
/ 2/x\\ / 2/x\\ / \ \2// | / \ \2// \2/|
|1 + cot |-||*|1 + cot |-|| * / 1 + --------------- *| / 1 + --------------- + ------------|
\ \2// \ \4// / 2 | / 2 2/x\ |
/ / 2/x\\ | / / 2/x\\ 1 + cot |-| |
/ |1 + cot |-|| | / |1 + cot |-|| \2/ |
\/ \ \2// \\/ \ \2// /
$$- \frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}{\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x\\
2 | 1 - tan |-||
/ 2/x\\ / 2/x\\ | \2/| /x\
-2*|1 - tan |-|| *|1 - tan |-||*|1 + -----------|*tan|-|
\ \4// \ \2// | 2/x\| \2/
| 1 + tan |-||
\ \2//
----------------------------------------------------------------------------------------------------------
____________________ / ____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
2 / |1 - tan |-|| | / |1 - tan |-|| 1 - tan |-||
/ 2/x\\ / 2/x\\ / \ \2// | / \ \2// \2/|
|1 + tan |-||*|1 + tan |-|| * / 1 + -------------- *| / 1 + -------------- + -----------|
\ \2// \ \4// / 2 | / 2 2/x\|
/ / 2/x\\ | / / 2/x\\ 1 + tan |-||
/ |1 + tan |-|| | / |1 + tan |-|| \2/|
\/ \ \2// \\/ \ \2// /
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}\right) \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ / pi\ / pi\\ |- cos|x - --| - cos(x)*cos|x - --||*cos(x) \ \ 2 / \ 2 // -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$\frac{\left(- \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(x - \frac{\pi}{2} \right)}\right) \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
-(1 + cos(x))*cos(x)*sin(x)
-------------------------------------
_____________
2 / 2
1 + cos (x) + \/ 1 + cos (x) *cos(x)
$$- \frac{\left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} + 1}$$
/ 2/x\ \
| cos |-| |
2/x pi\ | \2/ | / pi\
-cos |- - --|*(1 + cos(x))*|-1 + ------------|*cos|x - --|
\2 2 / | 2/x pi\| \ 2 /
| cos |- - --||
\ \2 2 //
-----------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ 2/x\ \
| sec |-| |
/ 1 \ | \2/ |
-|1 + ------|*|1 - ------------|
\ sec(x)/ | 2/x pi\|
| sec |- - --||
\ \2 2 //
--------------------------------------------------------------------
_____________ / _____________ \
/ 1 | / 1 1 | 2/x\ / pi\
/ 1 + ------- *| / 1 + ------- + ------|*sec |-|*sec|x - --|
/ 2 | / 2 sec(x)| \2/ \ 2 /
\/ sec (x) \\/ sec (x) /
$$- \frac{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} \left(\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
-sin(x)*sin(2*x)
------------------------------------------------
/ _____________ \
| 2 / 2 | /x\
2*\1 + cos (x) + \/ 1 + cos (x) *cos(x)/*tan|-|
\2/
$$- \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{2 \left(\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}$$
/ 2/x\\
| -1 + cot |-||
| \2/| / 2/x\\ /x\
-2*|1 + ------------|*|-1 + cot |-||*cot|-|
| 2/x\ | \ \2// \2/
| 1 + cot |-| |
\ \2/ /
-------------------------------------------------------------------------------------
/ _____________________ \
| / 2 |
| / / 2/x\\ |
| / |-1 + cot |-|| |
| / \ \2// / 2/x\\|
| / 1 + --------------- *|-1 + cot |-|||
| 2 / 2 \ \2//|
| / 2/x\\ / / 2/x\\ |
2 | |-1 + cot |-|| / |1 + cot |-|| |
/ 2/x\\ | \ \2// \/ \ \2// |
|1 + cot |-|| *|1 + --------------- + ----------------------------------------------|
\ \2// | 2 2/x\ |
| / 2/x\\ 1 + cot |-| |
| |1 + cot |-|| \2/ |
\ \ \2// /
$$- \frac{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\frac{\sqrt{\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1\right)}$$
2/x\ / 2/x\\
-sin |-|*(1 + cos(x))*|-1 + cot |-||*sin(x)
\2/ \ \2//
--------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(\cos{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ pi\
-(1 + cos(x))*cos(x)*cos|x - --|
\ 2 /
-------------------------------------
_____________
2 / 2
1 + cos (x) + \/ 1 + cos (x) *cos(x)
$$- \frac{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 1} \cos{\left(x \right)} + \cos^{2}{\left(x \right)} + 1}$$
2/x\ / 2/x\\
-cos |-|*|1 - tan |-||*(1 + cos(x))*sin(x)
\2/ \ \2//
--------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
/ 2/x\\
| 1 - tan |-||
/ 2/x\\ | \2/| /x\
-2*|1 - tan |-||*|1 + -----------|*tan|-|
\ \2// | 2/x\| \2/
| 1 + tan |-||
\ \2//
----------------------------------------------------------------------------------
/ ____________________ \
| / 2 |
| / / 2/x\\ |
| / |1 - tan |-|| |
| / \ \2// / 2/x\\|
| / 1 + -------------- *|1 - tan |-|||
| 2 / 2 \ \2//|
| / 2/x\\ / / 2/x\\ |
2 | |1 - tan |-|| / |1 + tan |-|| |
/ 2/x\\ | \ \2// \/ \ \2// |
|1 + tan |-|| *|1 + -------------- + --------------------------------------------|
\ \2// | 2 2/x\ |
| / 2/x\\ 1 + tan |-| |
| |1 + tan |-|| \2/ |
\ \ \2// /
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}$$
1 1
- ------ - -------------
csc(x) csc(x)*sec(x)
-------------------------------------------------------
_____________ / _____________ \
/ 1 | / 1 1 |
/ 1 + ------- *| / 1 + ------- + ------|*sec(x)
/ 2 | / 2 sec(x)|
\/ sec (x) \\/ sec (x) /
$$\frac{- \frac{1}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}}{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} \left(\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$
/ 2/x\ \
| csc |-| |
/ 1 \ | \2/ |
-|1 + -----------|*|-1 + ------------|
| /pi \| | 2/pi x\|
| csc|-- - x|| | csc |-- - -||
\ \2 // \ \2 2//
--------------------------------------------------------------------------------
__________________ / __________________ \
/ 1 | / 1 1 | 2/x\
/ 1 + ------------ *| / 1 + ------------ + -----------|*csc(x)*csc |-|
/ 2/pi \ | / 2/pi \ /pi \| \2/
/ csc |-- - x| | / csc |-- - x| csc|-- - x||
\/ \2 / \\/ \2 / \2 //
$$- \frac{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)}{\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} \left(\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)} \csc{\left(x \right)}}$$
-(2*sin(2*x) + sin(x) + sin(3*x)) ------------------------------------------------------------------- / __________________________ \ | 2 / 2 | 4*\(1 - cos(x)) + 2*cos(x) + \/ (1 - cos(x)) + 2*cos(x) *cos(x)/
$$- \frac{\sin{\left(x \right)} + 2 \sin{\left(2 x \right)} + \sin{\left(3 x \right)}}{4 \left(\left(1 - \cos{\left(x \right)}\right)^{2} + \sqrt{\left(1 - \cos{\left(x \right)}\right)^{2} + 2 \cos{\left(x \right)}} \cos{\left(x \right)} + 2 \cos{\left(x \right)}\right)}$$
/ / pi\\ / pi\
|-sin(x) - sin(x)*sin|x + --||*sin|x + --|
\ \ 2 // \ 2 /
-------------------------------------------------------------
__________________ / __________________ \
/ 2/ pi\ | / 2/ pi\ / pi\|
/ 1 + sin |x + --| *| / 1 + sin |x + --| + sin|x + --||
\/ \ 2 / \\/ \ 2 / \ 2 //
$$\frac{\left(- \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} - \sin{\left(x \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1} + \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1}}$$
4/x\ / 2/x\\ /x\
-2*cos |-|*|1 - tan |-||*(1 + cos(x))*tan|-|
\2/ \ \2// \2/
----------------------------------------------------
_____________
1 + cos(2*x) / 2 / 2/x\\
1 + ------------ + \/ 1 + cos (x) *|-1 + 2*cos |-||
2 \ \2//
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cos{\left(x \right)} + 1\right) \cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(2 \cos^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1} + \frac{\cos{\left(2 x \right)} + 1}{2} + 1}$$
2/x\ 2/x\ 4/x\ / 2/x\\ /x\
-8*cot |-|*sin |-|*sin |-|*(1 + cos(x))*|-1 + cot |-||*cot|-|
\4/ \2/ \4/ \ \2// \2/
-------------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{8 \left(\cos{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{4} \right)} \sin^{2}{\left(\frac{x}{2} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
2/x\
2*sin |-|*(1 + cos(x))*cos(x)*sin(x)
\2/
----------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *(-1 + cos(x))*\\/ 1 + cos (x) + cos(x)/
$$\frac{2 \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} - 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
1 1
- ------ - ------------------
csc(x) /pi \
csc(x)*csc|-- - x|
\2 /
-----------------------------------------------------------------------------
__________________ / __________________ \
/ 1 | / 1 1 | /pi \
/ 1 + ------------ *| / 1 + ------------ + -----------|*csc|-- - x|
/ 2/pi \ | / 2/pi \ /pi \| \2 /
/ csc |-- - x| | / csc |-- - x| csc|-- - x||
\/ \2 / \\/ \2 / \2 //
$$\frac{- \frac{1}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}}{\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} \left(\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ / pi\\ / pi\
-|1 + sin|x + --||*sin(x)*sin|x + --|
\ \ 2 // \ 2 /
-----------------------------------------------------
__________________
2/ pi\ / 2/ pi\ / pi\
1 + sin |x + --| + / 1 + sin |x + --| *sin|x + --|
\ 2 / \/ \ 2 / \ 2 /
$$- \frac{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1} \sin{\left(x + \frac{\pi}{2} \right)} + \sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1}$$
/ /x\ / 2/x\\ /x\\
| 2*tan|-| 2*|1 - tan |-||*tan|-||
/ 2/x\\ | \2/ \ \2// \2/|
|1 - tan |-||*|- ----------- - ----------------------|
\ \2// | 2/x\ 2 |
| 1 + tan |-| / 2/x\\ |
| \2/ |1 + tan |-|| |
\ \ \2// /
-------------------------------------------------------------------------------------------
____________________ / ____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
/ |1 - tan |-|| | / |1 - tan |-|| 1 - tan |-||
/ 2/x\\ / \ \2// | / \ \2// \2/|
|1 + tan |-||* / 1 + -------------- *| / 1 + -------------- + -----------|
\ \2// / 2 | / 2 2/x\|
/ / 2/x\\ | / / 2/x\\ 1 + tan |-||
/ |1 + tan |-|| | / |1 + tan |-|| \2/|
\/ \ \2// \\/ \ \2// /
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}\right) \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 4/x\\
| 4*sin |-||
2/pi x\ | \2/| / / pi\\
-sin |-- + -|*|1 - ---------|*|1 + sin|x + --||*sin(x)
\2 2/ | 2 | \ \ 2 //
\ sin (x) /
-------------------------------------------------------------
__________________ / __________________ \
/ 2/ pi\ | / 2/ pi\ / pi\|
/ 1 + sin |x + --| *| / 1 + sin |x + --| + sin|x + --||
\/ \ 2 / \\/ \ 2 / \ 2 //
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \sin{\left(x \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1} + \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin^{2}{\left(x + \frac{\pi}{2} \right)} + 1}}$$
/ 2/pi x\\
| csc |-- - -||
/ 1 \ | \2 2/|
-|1 + -----------|*|1 - ------------|
| /pi \| | 2/x\ |
| csc|-- - x|| | csc |-| |
\ \2 // \ \2/ /
-------------------------------------------------------------------------------------
__________________ / __________________ \
/ 1 | / 1 1 | 2/pi x\
/ 1 + ------------ *| / 1 + ------------ + -----------|*csc(x)*csc |-- - -|
/ 2/pi \ | / 2/pi \ /pi \| \2 2/
/ csc |-- - x| | / csc |-- - x| csc|-- - x||
\/ \2 / \\/ \2 / \2 //
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)}{\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} \left(\sqrt{1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ 2/x\\
| 1 - tan |-||
2/x\ | \2/| / 1 \ /x\
-8*tan |-|*|1 + -----------|*|-1 + -------|*tan|-|
\4/ | 2/x\| | 2/x\| \2/
| 1 + tan |-|| | tan |-||
\ \2// \ \2//
----------------------------------------------------------------------------------------------------------
____________________ / ____________________ \
/ 2 | / 2 |
/ / 2/x\\ | / / 2/x\\ 2/x\|
2 / |1 - tan |-|| | / |1 - tan |-|| 1 - tan |-||
/ 2/x\\ / 2/x\\ / \ \2// | / \ \2// \2/|
|1 + tan |-||*|1 + tan |-|| * / 1 + -------------- *| / 1 + -------------- + -----------|
\ \2// \ \4// / 2 | / 2 2/x\|
/ / 2/x\\ | / / 2/x\\ 1 + tan |-||
/ |1 + tan |-|| | / |1 + tan |-|| \2/|
\/ \ \2// \\/ \ \2// /
$$- \frac{8 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \tan^{2}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1}\right) \sqrt{\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2
/ 2/x\\ 2/x\ 4/x\ / 2/x\\ /x\
-2*|-1 + cot |-|| *sin |-|*sin |-|*|1 - tan |-||*(1 + cos(x))*cot|-|
\ \4// \2/ \4/ \ \2// \2/
--------------------------------------------------------------------
_____________ / _____________ \
/ 2 | / 2 |
\/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cos{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)} \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
4/x\ / 2/x\\ /x\
-2*sin |-|*(1 + cos(x))*|-1 + cot |-||*cot|-|
\2/ \ \2// \2/
----------------------------------------------------
_____________
1 + cos(2*x) / 2 / 2/x\\
1 + ------------ + \/ 1 + cos (x) *|-1 + 2*cos |-||
2 \ \2//
$$- \frac{2 \left(\cos{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(2 \cos^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt{\cos^{2}{\left(x \right)} + 1} + \frac{\cos{\left(2 x \right)} + 1}{2} + 1}$$
1 1
- ----------- - ------------------
/ pi\ / pi\
sec|x - --| sec(x)*sec|x - --|
\ 2 / \ 2 /
-------------------------------------------------------
_____________ / _____________ \
/ 1 | / 1 1 |
/ 1 + ------- *| / 1 + ------- + ------|*sec(x)
/ 2 | / 2 sec(x)|
\/ sec (x) \\/ sec (x) /
$$\frac{- \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}}{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} \left(\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$
(-sin(x) - cos(x)*sin(x))*cos(x) -------------------------------------------- _____________ / _____________ \ / 2 | / 2 | \/ 1 + cos (x) *\\/ 1 + cos (x) + cos(x)/
$$\frac{\left(- \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\sqrt{\cos^{2}{\left(x \right)} + 1} + \cos{\left(x \right)}\right) \sqrt{\cos^{2}{\left(x \right)} + 1}}$$
(-sin(x) - cos(x)*sin(x))*cos(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))