Mister Exam
Expression simplification/
Least common denominator/
sin(x)/(2*(1-cos(x))*sqrt(1+cos(x)))-(sqrt(2)-sqrt(1+cos(x)))*sin(x)/(1-cos(x))^2
Least common denominator sin(x)/(2*(1-cos(x))*sqrt(1+cos(x)))-(sqrt(2)-sqrt(1+cos(x)))*sin(x)/(1-cos(x))^2
The solution
You have entered
[src]
/ ___ ____________\
sin(x) \\/ 2 - \/ 1 + cos(x) /*sin(x)
----------------------------- - -------------------------------
____________ 2
2*(1 - cos(x))*\/ 1 + cos(x) (1 - cos(x))
$$- \frac{\left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} + \frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \sqrt{\cos{\left(x \right)} + 1}}$$
sin(x)/(((2*(1 - cos(x)))*sqrt(1 + cos(x)))) - (sqrt(2) - sqrt(1 + cos(x)))*sin(x)/(1 - cos(x))^2
General simplification
[src]
/ ______________ \
\3 - 2*\/ 2 + 2*cos(x) + cos(x)/*sin(x)
----------------------------------------
____________ 2
2*\/ 1 + cos(x) *(-1 + cos(x))
$$\frac{\left(- 2 \sqrt{2 \cos{\left(x \right)} + 2} + \cos{\left(x \right)} + 3\right) \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)^{2} \sqrt{\cos{\left(x \right)} + 1}}$$
(3 - 2*sqrt(2 + 2*cos(x)) + cos(x))*sin(x)/(2*sqrt(1 + cos(x))*(-1 + cos(x))^2)
Numerical answer
[src]
(1.0 + cos(x))^(-0.5)*sin(x)/(2.0 - 2.0*cos(x)) - (1.4142135623731 - (1.0 + cos(x))^0.5)*sin(x)/(1.0 - cos(x))^2
(1.0 + cos(x))^(-0.5)*sin(x)/(2.0 - 2.0*cos(x)) - (1.4142135623731 - (1.0 + cos(x))^0.5)*sin(x)/(1.0 - cos(x))^2
Combining rational expressions
[src]
/ ____________ / ___ ____________\\
\1 - cos(x) - 2*\/ 1 + cos(x) *\\/ 2 - \/ 1 + cos(x) //*sin(x)
---------------------------------------------------------------
2 ____________
2*(1 - cos(x)) *\/ 1 + cos(x)
$$\frac{\left(- 2 \left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \sqrt{\cos{\left(x \right)} + 1} - \cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)^{2} \sqrt{\cos{\left(x \right)} + 1}}$$
(1 - cos(x) - 2*sqrt(1 + cos(x))*(sqrt(2) - sqrt(1 + cos(x))))*sin(x)/(2*(1 - cos(x))^2*sqrt(1 + cos(x)))
Powers
[src]
/ __________________\
| / I*x -I*x |
| ___ / e e | / -I*x I*x\
I*|\/ 2 - / 1 + ---- + ----- |*\- e + e / / -I*x I*x\
\ \/ 2 2 / I*\- e + e /
---------------------------------------------------- - --------------------------------------------
2 __________________
/ I*x -I*x\ / I*x -I*x
| e e | / e e / I*x -I*x\
2*|1 - ---- - -----| 2* / 1 + ---- + ----- *\2 - e - e /
\ 2 2 / \/ 2 2
$$\frac{i \left(- \sqrt{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} + \sqrt{2}\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)^{2}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- e^{i x} + 2 - e^{- i x}\right) \sqrt{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}}}$$
/ ___ ____________\
sin(x) \\/ 2 - \/ 1 + cos(x) /*sin(x)
----------------------------- - -------------------------------
____________ 2
\/ 1 + cos(x) *(2 - 2*cos(x)) (1 - cos(x))
$$\frac{\sin{\left(x \right)}}{\left(2 - 2 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x \right)} + 1}} - \frac{\left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}$$
sin(x)/(sqrt(1 + cos(x))*(2 - 2*cos(x))) - (sqrt(2) - sqrt(1 + cos(x)))*sin(x)/(1 - cos(x))^2
Common denominator
[src]
___ ____________
3*sin(x) + cos(x)*sin(x) - 2*\/ 2 *\/ 1 + cos(x) *sin(x)
---------------------------------------------------------------------
____________ ____________ ____________ 2
2*\/ 1 + cos(x) - 4*\/ 1 + cos(x) *cos(x) + 2*\/ 1 + cos(x) *cos (x)
$$\frac{- 2 \sqrt{2} \sqrt{\cos{\left(x \right)} + 1} \sin{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)} + 1} \cos^{2}{\left(x \right)} - 4 \sqrt{\cos{\left(x \right)} + 1} \cos{\left(x \right)} + 2 \sqrt{\cos{\left(x \right)} + 1}}$$
(3*sin(x) + cos(x)*sin(x) - 2*sqrt(2)*sqrt(1 + cos(x))*sin(x))/(2*sqrt(1 + cos(x)) - 4*sqrt(1 + cos(x))*cos(x) + 2*sqrt(1 + cos(x))*cos(x)^2)
Rational denominator
[src]
2 2 ___ ____________ ___ ____________
-2*sin(x) - (-1 + cos(x)) *sin(x) + 2*cos (x)*sin(x) + 2*\/ 2 *\/ 1 + cos(x) *sin(x) - 2*\/ 2 *\/ 1 + cos(x) *cos(x)*sin(x)
---------------------------------------------------------------------------------------------------------------------------
____________ 2
\/ 1 + cos(x) *(-1 + cos(x)) *(-2 + 2*cos(x))
$$\frac{- \left(\cos{\left(x \right)} - 1\right)^{2} \sin{\left(x \right)} - 2 \sqrt{2} \sqrt{\cos{\left(x \right)} + 1} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \sqrt{2} \sqrt{\cos{\left(x \right)} + 1} \sin{\left(x \right)} + 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2} \sqrt{\cos{\left(x \right)} + 1} \left(2 \cos{\left(x \right)} - 2\right)}$$
(-2*sin(x) - (-1 + cos(x))^2*sin(x) + 2*cos(x)^2*sin(x) + 2*sqrt(2)*sqrt(1 + cos(x))*sin(x) - 2*sqrt(2)*sqrt(1 + cos(x))*cos(x)*sin(x))/(sqrt(1 + cos(x))*(-1 + cos(x))^2*(-2 + 2*cos(x)))
Combinatorics
[src]
/ ___ ____________ \
\3 - 2*\/ 2 *\/ 1 + cos(x) + cos(x)/*sin(x)
--------------------------------------------
____________ 2
2*\/ 1 + cos(x) *(-1 + cos(x))
$$\frac{\left(- 2 \sqrt{2} \sqrt{\cos{\left(x \right)} + 1} + \cos{\left(x \right)} + 3\right) \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)^{2} \sqrt{\cos{\left(x \right)} + 1}}$$
(3 - 2*sqrt(2)*sqrt(1 + cos(x)) + cos(x))*sin(x)/(2*sqrt(1 + cos(x))*(-1 + cos(x))^2)
Assemble expression
[src]
/ ___ ____________\
sin(x) \\/ 2 - \/ 1 + cos(x) /*sin(x)
----------------------------- - -------------------------------
____________ 2
\/ 1 + cos(x) *(2 - 2*cos(x)) (1 - cos(x))
$$\frac{\sin{\left(x \right)}}{\left(2 - 2 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x \right)} + 1}} - \frac{\left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}$$
sin(x)/(sqrt(1 + cos(x))*(2 - 2*cos(x))) - (sqrt(2) - sqrt(1 + cos(x)))*sin(x)/(1 - cos(x))^2
Trigonometric part
[src]
/ _________________\
| ___ / / pi\ |
|\/ 2 - / 1 + sin|x + --| |*sin(x)
sin(x) \ \/ \ 2 / /
----------------------------------------- - --------------------------------------
_________________ 2
/ / pi\ / / pi\\ / / pi\\
/ 1 + sin|x + --| *|2 - 2*sin|x + --|| |1 - sin|x + --||
\/ \ 2 / \ \ 2 // \ \ 2 //
$$\frac{\sin{\left(x \right)}}{\left(2 - 2 \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin{\left(x + \frac{\pi}{2} \right)} + 1}} - \frac{\left(- \sqrt{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \sqrt{2}\right) \sin{\left(x \right)}}{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}}$$
____________
___ / 1
\/ 2 - / 1 + ------
1 \/ sec(x)
----------------------------------------- - -------------------------
____________ 2
/ 1 / 2 \ / pi\ / 1 \ / pi\
/ 1 + ------ *|2 - ------|*sec|x - --| |1 - ------| *sec|x - --|
\/ sec(x) \ sec(x)/ \ 2 / \ sec(x)/ \ 2 /
$$\frac{1}{\sqrt{1 + \frac{1}{\sec{\left(x \right)}}} \left(2 - \frac{2}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{- \sqrt{1 + \frac{1}{\sec{\left(x \right)}}} + \sqrt{2}}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
sin(2*x) ___ 6*sin(x)
-------------- - 4*\/ 2 *sin(x) + --------------
____________ ____________
\/ 1 + cos(x) \/ 1 + cos(x)
------------------------------------------------
2
4*(-1 + cos(x))
$$\frac{- 4 \sqrt{2} \sin{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)} + 1}} + \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(x \right)} + 1}}}{4 \left(\cos{\left(x \right)} - 1\right)^{2}}$$
___ 3 1
- \/ 2 + ------------------ + -------------------------
____________ ____________
/ 1 / 1
2* / 1 + ------ 2* / 1 + ------ *sec(x)
\/ sec(x) \/ sec(x)
--------------------------------------------------------
2
/ 1 \ / pi\
|-1 + ------| *sec|x - --|
\ sec(x)/ \ 2 /
$$\frac{- \sqrt{2} + \frac{3}{2 \sqrt{1 + \frac{1}{\sec{\left(x \right)}}}} + \frac{1}{2 \sqrt{1 + \frac{1}{\sec{\left(x \right)}}} \sec{\left(x \right)}}}{\left(-1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
_________________
___ / 1
\/ 2 - / 1 + -----------
/ /pi \
/ csc|-- - x|
1 \/ \2 /
------------------------------------------------ - -------------------------------
_________________ 2
/ 1 / 2 \ / 1 \
/ 1 + ----------- *|2 - -----------|*csc(x) |1 - -----------| *csc(x)
/ /pi \ | /pi \| | /pi \|
/ csc|-- - x| | csc|-- - x|| | csc|-- - x||
\/ \2 / \ \2 // \ \2 //
$$\frac{1}{\sqrt{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \left(2 - \frac{2}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} - \frac{- \sqrt{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} + \sqrt{2}}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}}$$
/ __________________\
| / 2/x\ |
| / -1 + cot |-| |
| ___ / \2/ | /x\
2*|\/ 2 - / 1 + ------------ |*cot|-|
| / 2/x\ | \2/ /x\
| / 1 + cot |-| | 2*cot|-|
\ \/ \2/ / \2/
- --------------------------------------------- + ---------------------------------------------------------------
2 __________________
/ 2/x\\ / 2/x\ / / 2/x\\\
| -1 + cot |-|| / -1 + cot |-| | 2*|-1 + cot |-|||
/ 2/x\\ | \2/| / 2/x\\ / \2/ | \ \2//|
|1 + cot |-||*|1 - ------------| |1 + cot |-||* / 1 + ------------ *|2 - ----------------|
\ \2// | 2/x\ | \ \2// / 2/x\ | 2/x\ |
| 1 + cot |-| | / 1 + cot |-| | 1 + cot |-| |
\ \2/ / \/ \2/ \ \2/ /
$$- \frac{2 \left(- \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1} + \sqrt{2}\right) \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x\ \
| 1 - tan |-| |
| ___ 3 \2/ | /x\
2*|- \/ 2 + --------------------------- + -----------------------------------------|*tan|-|
| _________________ _________________| \2/
| / 2/x\ / 2/x\ |
| / 1 - tan |-| / 1 - tan |-| |
| / \2/ / 2/x\\ / \2/ |
| 2* / 1 + ----------- 2*|1 + tan |-||* / 1 + ----------- |
| / 2/x\ \ \2// / 2/x\ |
| / 1 + tan |-| / 1 + tan |-| |
\ \/ \2/ \/ \2/ /
--------------------------------------------------------------------------------------------
2
/ 2/x\\
| 1 - tan |-||
/ 2/x\\ | \2/|
|1 + tan |-||*|-1 + -----------|
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 \sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \sqrt{2} + \frac{3}{2 \sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1}}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ / pi\ \
| sin|x + --| |
| ___ 3 \ 2 / |
|- \/ 2 + ----------------------- + -----------------------|*sin(x)
| _________________ _________________|
| / / pi\ / / pi\ |
| 2* / 1 + sin|x + --| 2* / 1 + sin|x + --| |
\ \/ \ 2 / \/ \ 2 / /
--------------------------------------------------------------------
2
/ / pi\\
|-1 + sin|x + --||
\ \ 2 //
$$\frac{\left(- \sqrt{2} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \sqrt{\sin{\left(x + \frac{\pi}{2} \right)} + 1}} + \frac{3}{2 \sqrt{\sin{\left(x + \frac{\pi}{2} \right)} + 1}}\right) \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} - 1\right)^{2}}$$
/ ___ 3 cos(x) \
|- \/ 2 + ---------------- + ----------------|*sin(x)
| ____________ ____________|
\ 2*\/ 1 + cos(x) 2*\/ 1 + cos(x) /
------------------------------------------------------
2
(-1 + cos(x))
$$\frac{\left(- \sqrt{2} + \frac{\cos{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)} + 1}} + \frac{3}{2 \sqrt{\cos{\left(x \right)} + 1}}\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}$$
____________
___ / 1
\/ 2 - / 1 + ------
1 \/ sec(x)
------------------------------------ - ------------------------
____________ 2
/ 1 / 2 \ / 1 \
/ 1 + ------ *|2 - ------|*csc(x) |1 - ------| *csc(x)
\/ sec(x) \ sec(x)/ \ sec(x)/
$$\frac{1}{\sqrt{1 + \frac{1}{\sec{\left(x \right)}}} \left(2 - \frac{2}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{- \sqrt{1 + \frac{1}{\sec{\left(x \right)}}} + \sqrt{2}}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}}$$
/ _________________\
| / 2/x\ |
| / 1 - tan |-| |
| ___ / \2/ | /x\
2*|\/ 2 - / 1 + ----------- |*tan|-|
| / 2/x\ | \2/ /x\
| / 1 + tan |-| | 2*tan|-|
\ \/ \2/ / \2/
- -------------------------------------------- + -------------------------------------------------------------
2 _________________
/ 2/x\\ / 2/x\ / / 2/x\\\
| 1 - tan |-|| / 1 - tan |-| | 2*|1 - tan |-|||
/ 2/x\\ | \2/| / 2/x\\ / \2/ | \ \2//|
|1 + tan |-||*|1 - -----------| |1 + tan |-||* / 1 + ----------- *|2 - ---------------|
\ \2// | 2/x\| \ \2// / 2/x\ | 2/x\ |
| 1 + tan |-|| / 1 + tan |-| | 1 + tan |-| |
\ \2// \/ \2/ \ \2/ /
$$- \frac{2 \left(- \sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1} + \sqrt{2}\right) \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 2\right) \sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ pi\ / ___ ____________\ / pi\
cos|x - --| \\/ 2 - \/ 1 + cos(x) /*cos|x - --|
\ 2 / \ 2 /
----------------------------- - ------------------------------------
____________ 2
\/ 1 + cos(x) *(2 - 2*cos(x)) (1 - cos(x))
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(2 - 2 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x \right)} + 1}} - \frac{\left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}$$
___ 3 1
- \/ 2 + ------------------------- + -------------------------------------
_________________ _________________
/ 1 / 1 /pi \
2* / 1 + ----------- 2* / 1 + ----------- *csc|-- - x|
/ /pi \ / /pi \ \2 /
/ csc|-- - x| / csc|-- - x|
\/ \2 / \/ \2 /
---------------------------------------------------------------------------
2
/ 1 \
|-1 + -----------| *csc(x)
| /pi \|
| csc|-- - x||
\ \2 //
$$\frac{- \sqrt{2} + \frac{3}{2 \sqrt{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}} + \frac{1}{2 \sqrt{1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(-1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}}$$
/ ___ ____________\
sin(x) \\/ 2 - \/ 1 + cos(x) /*sin(x)
----------------------------- - -------------------------------
____________ 2
\/ 1 + cos(x) *(2 - 2*cos(x)) (1 - cos(x))
$$\frac{\sin{\left(x \right)}}{\left(2 - 2 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x \right)} + 1}} - \frac{\left(- \sqrt{\cos{\left(x \right)} + 1} + \sqrt{2}\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}}$$
/ ___ 3 cos(x) \ / pi\
|- \/ 2 + ---------------- + ----------------|*cos|x - --|
| ____________ ____________| \ 2 /
\ 2*\/ 1 + cos(x) 2*\/ 1 + cos(x) /
-----------------------------------------------------------
2
(-1 + cos(x))
$$\frac{\left(- \sqrt{2} + \frac{\cos{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)} + 1}} + \frac{3}{2 \sqrt{\cos{\left(x \right)} + 1}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}$$
/ 2/x\ \
| -1 + cot |-| |
| ___ 3 \2/ | /x\
2*|- \/ 2 + ---------------------------- + ------------------------------------------|*cot|-|
| __________________ __________________| \2/
| / 2/x\ / 2/x\ |
| / -1 + cot |-| / -1 + cot |-| |
| / \2/ / 2/x\\ / \2/ |
| 2* / 1 + ------------ 2*|1 + cot |-||* / 1 + ------------ |
| / 2/x\ \ \2// / 2/x\ |
| / 1 + cot |-| / 1 + cot |-| |
\ \/ \2/ \/ \2/ /
----------------------------------------------------------------------------------------------
2
/ 2/x\\
| -1 + cot |-||
/ 2/x\\ | \2/|
|1 + cot |-||*|-1 + ------------|
\ \2// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{2 \left(- \sqrt{2} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{3}{2 \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1}}\right) \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2*(-sqrt(2) + 3/(2*sqrt(1 + (-1 + cot(x/2)^2)/(1 + cot(x/2)^2))) + (-1 + cot(x/2)^2)/(2*(1 + cot(x/2)^2)*sqrt(1 + (-1 + cot(x/2)^2)/(1 + cot(x/2)^2))))*cot(x/2)/((1 + cot(x/2)^2)*(-1 + (-1 + cot(x/2)^2)/(1 + cot(x/2)^2))^2)