Mister Exam
Least common denominator cos(x)/sin(x)^3+(1+tan(x)^2)/tan(x)
The solution
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2 cos(x) 1 + tan (x) ------- + ----------- 3 tan(x) sin (x)
$$\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$
cos(x)/sin(x)^3 + (1 + tan(x)^2)/tan(x)
General simplification
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1
--------------
3
cos(x)*sin (x)
$$\frac{1}{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}$$
1/(cos(x)*sin(x)^3)
Combining rational expressions
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3 / 2 \
sin (x)*\1 + tan (x)/ + cos(x)*tan(x)
-------------------------------------
3
sin (x)*tan(x)
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{3}{\left(x \right)} \tan{\left(x \right)}}$$
(sin(x)^3*(1 + tan(x)^2) + cos(x)*tan(x))/(sin(x)^3*tan(x))
Combinatorics
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3 3 2
sin (x) + sin (x)*tan (x) + cos(x)*tan(x)
-----------------------------------------
3
sin (x)*tan(x)
$$\frac{\sin^{3}{\left(x \right)} \tan^{2}{\left(x \right)} + \sin^{3}{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{3}{\left(x \right)} \tan{\left(x \right)}}$$
(sin(x)^3 + sin(x)^3*tan(x)^2 + cos(x)*tan(x))/(sin(x)^3*tan(x))
Rational denominator
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3 / 2 \
sin (x)*\1 + tan (x)/ + cos(x)*tan(x)
-------------------------------------
3
sin (x)*tan(x)
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{3}{\left(x \right)} \tan{\left(x \right)}}$$
(sin(x)^3*(1 + tan(x)^2) + cos(x)*tan(x))/(sin(x)^3*tan(x))
Numerical answer
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cos(x)/sin(x)^3 + (1.0 + tan(x)^2)/tan(x)
cos(x)/sin(x)^3 + (1.0 + tan(x)^2)/tan(x)
Common denominator
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3
sin (x) + cos(x)*tan(x)
----------------------- + tan(x)
3
sin (x)*tan(x)
$$\frac{\sin^{3}{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}}{\sin^{3}{\left(x \right)} \tan{\left(x \right)}} + \tan{\left(x \right)}$$
(sin(x)^3 + cos(x)*tan(x))/(sin(x)^3*tan(x)) + tan(x)
Powers
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/ 2\
| / I*x -I*x\ |
| \- e + e / | / I*x -I*x\
/ I*x -I*x\ I*|1 - -----------------|*\e + e /
|e e | | 2 |
8*I*|---- + -----| | / I*x -I*x\ |
\ 2 2 / \ \e + e / /
- ------------------ - ----------------------------------------
3 I*x -I*x
/ -I*x I*x\ - e + e
\- e + e /
$$- \frac{i \left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right)}{- e^{i x} + e^{- i x}} - \frac{8 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{\left(e^{i x} - e^{- i x}\right)^{3}}$$
-8*i*(exp(i*x)/2 + exp(-i*x)/2)/(-exp(-i*x) + exp(i*x))^3 - i*(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))/(-exp(i*x) + exp(-i*x))
Trigonometric part
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4
/ 2/x\\ / 2/x\\
2 |1 + tan |-|| *|1 - tan |-||
1 + tan (x) \ \4// \ \2//
----------- + ----------------------------
tan(x) 4
/ 2/x\\ 3/x\
8*|1 - tan |-|| *tan |-|
\ \4// \2/
$$\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \tan^{3}{\left(\frac{x}{2} \right)}}$$
/ 2/ pi\\
| cos |x - --||
| \ 2 /|
|1 + ------------|*cos(x)
| 2 |
cos(x) \ cos (x) /
------------ + -------------------------
3/ pi\ / pi\
cos |x - --| cos|x - --|
\ 2 / \ 2 /
$$\frac{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x \right)}}{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}$$
/ 4 \
| 4*sin (x)|
|1 + ---------|*sin(2*x)
| 2 |
cos(x) \ sin (2*x)/
------- + ------------------------
3 2
sin (x) 2*sin (x)
$$\frac{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$
-8 ---------------------- -2*sin(2*x) + sin(4*x)
$$- \frac{8}{- 2 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}$$
/ 2 \
| sin (x)|
|1 + -------|*cos(x)
| 2 |
cos(x) \ cos (x)/
------- + --------------------
3 sin(x)
sin (x)
$$\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$
3 csc (x) ------- cos(x)
$$\frac{\csc^{3}{\left(x \right)}}{\cos{\left(x \right)}}$$
4
/ 2/x\\ / 2/x\\
2 |1 + cot |-|| *|-1 + cot |-||
1 + cot (x) \ \4// \ \2//
----------- + -----------------------------
cot(x) 3/x\ 4/x\
128*cot |-|*cot |-|
\2/ \4/
$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{128 \cot^{4}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}} + \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}$$
4
/ 2/x\\ 3/x\ / 1 \
|1 + cot |-|| *cot |-|*|1 - -------|
\ \4// \2/ | 2/x\|
2 | cot |-||
1 + cot (x) \ \2//
----------- + ------------------------------------
cot(x) 4
/ 2/x\\
8*|-1 + cot |-||
\ \4//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{3}{\left(\frac{x}{2} \right)}}{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}} + \frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}}$$
2
/ 2/x\\ / 2/x\\
|1 + cot |-|| *|-1 + cot |-||
/ 1 \ \ \2// \ \2//
|1 + -------|*cot(x) + -----------------------------
| 2 | 3/x\
\ cot (x)/ 8*cot |-|
\2/
$$\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{8 \cot^{3}{\left(\frac{x}{2} \right)}}$$
2/x\
1 - tan |-|
2 \2/
-------- + -----------------
sin(2*x) 4/x\ 3/x\
8*cos |-|*tan |-|
\2/ \2/
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{8 \cos^{4}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
2/x\
1 - tan |-|
2 \2/
-------- + --------------------------------
sin(2*x) 4
/ 2/x\\ 8/x\ 3/x\
8*|1 - tan |-|| *cos |-|*tan |-|
\ \4// \4/ \2/
$$\frac{2}{\sin{\left(2 x \right)}} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}$$
3/ pi\
sec |x - --|*sec(x)
\ 2 /
$$\sec{\left(x \right)} \sec^{3}{\left(x - \frac{\pi}{2} \right)}$$
1 ------------------------------- / 2/x\\ 8/x\ 3/x\ 8*|1 - tan |-||*cos |-|*tan |-| \ \2// \2/ \2/
$$\frac{1}{8 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{8}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}$$
/ 1 \ /x\
|-1 + -------|*csc|-|
| 2/x\| \2/
| tan |-||
\ \2//
2*csc(2*x) + ---------------------
3/x\
8*cos |-|
\2/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{8 \cos^{3}{\left(\frac{x}{2} \right)}} + 2 \csc{\left(2 x \right)}$$
4
/ 2/x\\
|1 + tan |-||
\ \2//
-----------------------
/ 2/x\\ 3/x\
8*|1 - tan |-||*tan |-|
\ \2// \2/
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{8 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan^{3}{\left(\frac{x}{2} \right)}}$$
2/x pi\
cos |- - --|
\2 2 /
1 - ------------
2/x\
cos |-|
2 \2/
------------- + ---------------------
/ pi\ /x\ 3/x pi\
cos|2*x - --| 8*cos|-|*cos |- - --|
\ 2 / \2/ \2 2 /
$$\frac{1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}}{8 \cos{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2}{\cos{\left(2 x - \frac{\pi}{2} \right)}}$$
sec(x) ------- 3 sin (x)
$$\frac{\sec{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$
2/x\
cos |-|
\2/
-1 + ------------
2/x pi\
cos |- - --|
2 \2 2 /
------------- + ---------------------
/ pi\ 3/x\ /x pi\
cos|2*x - --| 8*cos |-|*cos|- - --|
\ 2 / \2/ \2 2 /
$$\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{8 \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2}{\cos{\left(2 x - \frac{\pi}{2} \right)}}$$
/ 2 \
2/x\ | sin (x) |
sin |-|*|-1 + ---------|
\2/ | 4/x\|
| 4*sin |-||
2 \ \2//
-------- + ------------------------
sin(2*x) 3
sin (x)
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\sin^{3}{\left(x \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
/ 2/pi \\
| csc |-- - x||
| \2 /|
|1 + ------------|*csc(x)
3 | 2 |
csc (x) \ csc (x) /
----------- + -------------------------
/pi \ /pi \
csc|-- - x| csc|-- - x|
\2 / \2 /
$$\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc^{3}{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$
1
--------------
3
cos(x)*sin (x)
$$\frac{1}{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}$$
/ 2/pi x\\
| csc |-- - -||
3/x\ | \2 2/| /pi x\
csc |-|*|1 - ------------|*csc|-- - -|
\2/ | 2/x\ | \2 2/
| csc |-| |
\ \2/ /
2*csc(2*x) + --------------------------------------
8
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc^{3}{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{8} + 2 \csc{\left(2 x \right)}$$
/ 2/x\\ /x\
|1 - tan |-||*sec|-|
2 \ \2// \2/
-------- + --------------------
sin(2*x) 3/x\
8*sin |-|
\2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sec{\left(\frac{x}{2} \right)}}{8 \sin^{3}{\left(\frac{x}{2} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
/ 2 \
| sec (x)|
|1 + -------|*csc(x)
3 | 2 |
csc (x) \ csc (x)/
------- + --------------------
sec(x) sec(x)
$$\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{\csc^{3}{\left(x \right)}}{\sec{\left(x \right)}}$$
/ 2 \
| sec (x) | / pi\
|1 + ------------|*sec|x - --|
3/ pi\ | 2/ pi\| \ 2 /
sec |x - --| | sec |x - --||
\ 2 / \ \ 2 //
------------ + ------------------------------
sec(x) sec(x)
$$\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} + \frac{\sec^{3}{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}$$
/ 2/x\\ /x\
|1 - tan |-||*tan|-|
2 \ \2// \2/
-------- + --------------------
sin(2*x) 4/x\ 4/x\
128*cos |-|*sin |-|
\4/ \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{128 \sin^{4}{\left(\frac{x}{4} \right)} \cos^{4}{\left(\frac{x}{4} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
/ 2/x\ \
| sec |-| |
3/x pi\ | \2/ | /x\
sec |- - --|*|1 - ------------|*sec|-|
\2 2 / | 2/x pi\| \2/
| sec |- - --||
/ pi\ \ \2 2 //
2*sec|2*x - --| + --------------------------------------
\ 2 / 8
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{8} + 2 \sec{\left(2 x - \frac{\pi}{2} \right)}$$
3 /pi \
csc (x)*csc|-- - x|
\2 /
$$\csc^{3}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}$$
4
/ 2/x\\
|1 + cot |-||
\ \2//
------------------------
/ 2/x\\ 3/x\
8*|-1 + cot |-||*cot |-|
\ \2// \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}}{8 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot^{3}{\left(\frac{x}{2} \right)}}$$
/ 4/x\\
| 4*sin |-||
3 | \2/|
sin (x)*|1 - ---------|
| 2 |
2 \ sin (x) /
-------- + -----------------------
sin(2*x) 6/x\ 4/pi x\
64*sin |-|*sin |-- + -|
\2/ \2 2/
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{3}{\left(x \right)}}{64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
3/x\ / 2/x\\
tan |-|*|-1 + cot |-||
2 \2/ \ \2//
-------- + ----------------------
sin(2*x) 4/x\
8*sin |-|
\2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \tan^{3}{\left(\frac{x}{2} \right)}}{8 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
4
/ 2/x\\ 3/x\ / 1 \
|1 + tan |-|| *tan |-|*|-1 + -------|
\ \4// \2/ | 2/x\|
2 | tan |-||
1 + tan (x) \ \2//
----------- + -------------------------------------
tan(x) 4/x\
128*tan |-|
\4/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{3}{\left(\frac{x}{2} \right)}}{128 \tan^{4}{\left(\frac{x}{4} \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}$$
/ 2/x pi\\
| sec |- - --||
3/x\ | \2 2 /| /x pi\
sec |-|*|-1 + ------------|*sec|- - --|
\2/ | 2/x\ | \2 2 /
| sec |-| |
/ pi\ \ \2/ /
2*sec|2*x - --| + ---------------------------------------
\ 2 / 8
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{8} + 2 \sec{\left(2 x - \frac{\pi}{2} \right)}$$
2/x\
sin |-|
1 1 \2/
------ + --------- - -------
cos(x) 2/x\ 2
4*sin |-| sin (x)
\2/
----------------------------
sin(x)
$$\frac{- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{\cos{\left(x \right)}} + \frac{1}{4 \sin^{2}{\left(\frac{x}{2} \right)}}}{\sin{\left(x \right)}}$$
1
-------------------
3 / pi\
sin (x)*sin|x + --|
\ 2 /
$$\frac{1}{\sin^{3}{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}$$
/ 2/x\ \
| csc |-| |
3/pi x\ | \2/ | /x\
csc |-- - -|*|-1 + ------------|*csc|-|
\2 2/ | 2/pi x\| \2/
| csc |-- - -||
\ \2 2//
2*csc(2*x) + ---------------------------------------
8
$$\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{8} + 2 \csc{\left(2 x \right)}$$
1
-------------------
3/ pi\
cos(x)*cos |x - --|
\ 2 /
$$\frac{1}{\cos{\left(x \right)} \cos^{3}{\left(x - \frac{\pi}{2} \right)}}$$
/ 4 \
| 4*sin (x)|
/ pi\ |1 + ---------|*sin(2*x)
sin|x + --| | 2 |
\ 2 / \ sin (2*x)/
----------- + ------------------------
3 2
sin (x) 2*sin (x)
$$\frac{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\sin^{3}{\left(x \right)}}$$
3/x\ / 2/x\\
cot |-|*|1 - tan |-||
2 \2/ \ \2//
-------- + -------------------------
sin(2*x) 4
/ 2/x\\ 8/x\
8*|-1 + cot |-|| *sin |-|
\ \4// \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cot^{3}{\left(\frac{x}{2} \right)}}{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}} + \frac{2}{\sin{\left(2 x \right)}}$$
2
/ 2/x\\ / 2/x\\
2 |1 + tan |-|| *|1 - tan |-||
1 + tan (x) \ \2// \ \2//
----------- + ----------------------------
tan(x) 3/x\
8*tan |-|
\2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{8 \tan^{3}{\left(\frac{x}{2} \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}}$$
3/x\
tan |-|
\2/
------------------------
/ 2/x\\ 8/x\
8*|-1 + cot |-||*sin |-|
\ \2// \2/
$$\frac{\tan^{3}{\left(\frac{x}{2} \right)}}{8 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{8}{\left(\frac{x}{2} \right)}}$$
tan(x/2)^3/(8*(-1 + cot(x/2)^2)*sin(x/2)^8)