Mister Exam
Expression simplification/
Least common denominator/
cos(3*pi/2+x)*cos(x-pi)-sin(x-(pi/2))^(2)-sin(2*pi-x)^(2)
Least common denominator cos(3*pi/2+x)*cos(x-pi)-sin(x-(pi/2))^(2)-sin(2*pi-x)^(2)
The solution
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/3*pi \ 2/ pi\ 2 cos|---- + x|*cos(x - pi) - sin |x - --| - sin (2*pi - x) \ 2 / \ 2 /
$$\left(- \sin^{2}{\left(x - \frac{\pi}{2} \right)} + \cos{\left(x - \pi \right)} \cos{\left(x + \frac{3 \pi}{2} \right)}\right) - \sin^{2}{\left(- x + 2 \pi \right)}$$
cos((3*pi)/2 + x)*cos(x - pi) - sin(x - pi/2)^2 - sin(2*pi - x)^2
General simplification
[src]
sin(2*x)
-1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} - 1$$
-1 - sin(2*x)/2
Powers
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2
/ /pi \ / pi\\ / / 3*pi\ / 3*pi\\
2 | I*|-- - x| I*|x - --|| | I*|x + ----| I*|-x - ----||
/ I*(x - 2*pi) I*(-x + 2*pi)\ | \2 / \ 2 /| / I*(pi - x) I*(x - pi)\ | \ 2 / \ 2 /|
\- e + e / \- e + e / |e e | |e e |
----------------------------------- + ------------------------------ + |----------- + -----------|*|------------- + --------------|
4 4 \ 2 2 / \ 2 2 /
$$\left(\frac{e^{i \left(\pi - x\right)}}{2} + \frac{e^{i \left(x - \pi\right)}}{2}\right) \left(\frac{e^{i \left(- x - \frac{3 \pi}{2}\right)}}{2} + \frac{e^{i \left(x + \frac{3 \pi}{2}\right)}}{2}\right) + \frac{\left(- e^{i \left(- x + \frac{\pi}{2}\right)} + e^{i \left(x - \frac{\pi}{2}\right)}\right)^{2}}{4} + \frac{\left(e^{i \left(- x + 2 \pi\right)} - e^{i \left(x - 2 \pi\right)}\right)^{2}}{4}$$
2 2 - cos (x) - sin (x) - cos(x)*sin(x)
$$- \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}$$
-cos(x)^2 - sin(x)^2 - cos(x)*sin(x)
Numerical answer
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-sin(x - pi/2)^2 - sin(2*pi - x)^2 + cos(x - pi)*cos((3*pi)/2 + x)
-sin(x - pi/2)^2 - sin(2*pi - x)^2 + cos(x - pi)*cos((3*pi)/2 + x)
Combining rational expressions
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2 2/-pi + 2*x\ /2*x + 3*pi\
- sin (x) - sin |---------| - cos(x)*cos|----------|
\ 2 / \ 2 /
$$- \sin^{2}{\left(x \right)} - \sin^{2}{\left(\frac{2 x - \pi}{2} \right)} - \cos{\left(x \right)} \cos{\left(\frac{2 x + 3 \pi}{2} \right)}$$
-sin(x)^2 - sin((-pi + 2*x)/2)^2 - cos(x)*cos((2*x + 3*pi)/2)
Rational denominator
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2 2 - cos (x) - sin (x) - cos(x)*sin(x)
$$- \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}$$
-cos(x)^2 - sin(x)^2 - cos(x)*sin(x)
Combinatorics
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2 2 - cos (x) - sin (x) - cos(x)*sin(x)
$$- \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}$$
-cos(x)^2 - sin(x)^2 - cos(x)*sin(x)
Common denominator
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2 2 - cos (x) - sin (x) + cos(x - pi)*sin(x)
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x - \pi \right)} - \cos^{2}{\left(x \right)}$$
-cos(x)^2 - sin(x)^2 + cos(x - pi)*sin(x)
Assemble expression
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2/ pi\ 2 /3*pi \
- sin |x - --| - sin (2*pi - x) + cos(x - pi)*cos|---- + x|
\ 2 / \ 2 /
$$- \sin^{2}{\left(- x + 2 \pi \right)} - \sin^{2}{\left(x - \frac{\pi}{2} \right)} + \cos{\left(x - \pi \right)} \cos{\left(x + \frac{3 \pi}{2} \right)}$$
-sin(x - pi/2)^2 - sin(2*pi - x)^2 + cos(x - pi)*cos((3*pi)/2 + x)
Trigonometric part
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2 2/ 3*pi\
- cos (x - pi) - cos |-x + ----| - cos(x)*sin(x)
\ 2 /
$$- \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(- x + \frac{3 \pi}{2} \right)} - \cos^{2}{\left(x - \pi \right)}$$
2 2 / pi\
- cos (x) - sin (x) + sin(x + 2*pi)*sin|x - --|
\ 2 /
$$- \sin^{2}{\left(x \right)} + \sin{\left(x - \frac{\pi}{2} \right)} \sin{\left(x + 2 \pi \right)} - \cos^{2}{\left(x \right)}$$
/ 2\
| / 2/x\\ |
| 2*|-1 + cot |-|| |
| \ \4// | /x\
2*|1 - -----------------|*cot|-|
| 2 | \2/
| / 2/x\\ |
2 | |1 + cot |-|| |
1 -1 + cot (x) 4 \ \ \4// /
- - + --------------- - -------------------------------- + --------------------------------
2 / 2 \ 2 2/x\
2*\1 + cot (x)/ / 1 \ 2/x pi\ 1 + cot |-|
|1 + ------------| *cot |- + --| \2/
| 2/x pi\| \2 4 /
| cot |- + --||
\ \2 4 //
$$\frac{2 \left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + 1\right) \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{\cot^{2}{\left(x \right)} - 1}{2 \left(\cot^{2}{\left(x \right)} + 1\right)} - \frac{1}{2} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
2
/ /x\ \ / /x\ \
| 2*tan|-| | | 2*tan|-| |
| \2/ | 2/x\ / 1 \ / 1 \ | \2/ |
|1 + -----------| 2*tan |-|*|1 - -------|*|1 - ------------|*|1 + -----------|
| 2/x\| \4/ | 2/x\| | 2/x pi\| | 2/x\|
2 | 1 + tan |-|| | tan |-|| | tan |- + --|| | 1 + tan |-||
1 1 - tan (x) \ \2// \ \2// \ \2 4 // \ \2//
- - + --------------- - ------------------ + ------------------------------------------------------------
2 / 2 \ 2/x pi\ 2
2*\1 + tan (x)/ tan |- + --| / 2/x\\
\2 4 / |1 + tan |-||
\ \4//
$$- \frac{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + \frac{2 \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{1 - \tan^{2}{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{1}{2}$$
1 - cos(2*x) / 2/x\\ 4
- ------------ + |1 - 2*cos |-||*sin(x) - --------------------------------
2 \ \2// 2
/ 1 \ 2/x pi\
|1 + ------------| *tan |- + --|
| 2/x pi\| \2 4 /
| tan |- + --||
\ \2 4 //
$$\left(1 - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(x \right)} - \frac{1 - \cos{\left(2 x \right)}}{2} - \frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
1 1 1
- ------------ - --------------- - -------------
2 2/ 3*pi\ csc(x)*sec(x)
sec (x - pi) sec |-x + ----|
\ 2 /
$$- \frac{1}{\sec^{2}{\left(x - \pi \right)}} - \frac{1}{\sec^{2}{\left(- x + \frac{3 \pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}$$
1 1 1
- ------- - ------- - ------------------
2 2 /pi \
csc (x) sec (x) csc(x)*csc|-- - x|
\2 /
$$- \frac{1}{\sec^{2}{\left(x \right)}} - \frac{1}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)}}$$
tan(x)
-1 - -----------
2
1 + tan (x)
$$-1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
1 cos(2*x) 2 2/x pi\ 2/x\ 4/x\ / 2/x\\ / 2/x pi\\ - - + -------- - (1 + sin(x)) *cot |- + --| + 2*cot |-|*sin |-|*|1 - cot |-||*|1 - cot |- + --||*(1 + sin(x)) 2 2 \2 4 / \4/ \4/ \ \2// \ \2 4 //
$$2 \left(1 - \cot^{2}{\left(\frac{x}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right) \left(\sin{\left(x \right)} + 1\right) \sin^{4}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{4} \right)} - \left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2}$$
/ 1 \ / 1 \
|-1 + -------|*|-1 + ------------|
2/x\ | 2/x\| | 2/x pi\|
4*cot |-| | cot |-|| | cot |- + --||
\2/ 4 \ \2// \ \2 4 //
- -------------- - -------------------------------- + ----------------------------------
2 2 / 1 \ / 1 \
/ 2/x\\ / 1 \ 2/x pi\ |1 + -------|*|1 + ------------|
|1 + cot |-|| |1 + ------------| *cot |- + --| | 2/x\| | 2/x pi\|
\ \2// | 2/x pi\| \2 4 / | cot |-|| | cot |- + --||
| cot |- + --|| \ \2// \ \2 4 //
\ \2 4 //
$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(-1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)} - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
2 2 / pi\
- cos (x) - sin (x) - cos(x)*cos|x - --|
\ 2 /
$$- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} - \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$
/ /x\ \
| 2*cot|-| |
2/x\ / 2/x\\ / 2/x pi\\ | \2/ |
2 2*cot |-|*|1 - cot |-||*|1 - cot |- + --||*|1 + -----------|
/ /x\ \ \4/ \ \2// \ \2 4 // | 2/x\|
2 | 2*cot|-| | | 1 + cot |-||
1 -1 + cot (x) | \2/ | 2/x pi\ \ \2//
- - + --------------- - |1 + -----------| *cot |- + --| + ------------------------------------------------------------
2 / 2 \ | 2/x\| \2 4 / 2
2*\1 + cot (x)/ | 1 + cot |-|| / 2/x\\
\ \2// |1 + cot |-||
\ \4//
$$- \left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + \frac{2 \left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right) \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(x \right)} - 1}{2 \left(\cot^{2}{\left(x \right)} + 1\right)} - \frac{1}{2}$$
/ pi\
cos|2*x - --|
\ 2 /
-1 - -------------
2
$$- \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2} - 1$$
1 1 1
- ------- - ------- - ------------------
2 2 /pi \
csc (x) sec (x) sec(x)*sec|-- - x|
\2 /
$$- \frac{1}{\sec{\left(x \right)} \sec{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)}}$$
/ 2/x pi\\ / 2/x pi\\
2 | sec |- - --|| | sec |- - --||
/ 1 \ 2/x pi\ / 1 \ | \2 2 /| | \2 4 /|
|1 + -----------| *sec |- - --| |1 + -----------|*|1 - ------------|*|1 - ------------|
| / pi\| \2 4 / | / pi\| | 2/x\ | | 2/x pi\|
| sec|x - --|| | sec|x - --|| | sec |-| | | sec |- + --||
1 1 \ \ 2 // \ \ 2 // \ \2/ / \ \2 4 //
- - + ---------- - ------------------------------- + -------------------------------------------------------
2 2*sec(2*x) 2/x pi\ 2/x pi\
sec |- + --| 2*sec |- - --|
\2 4 / \2 2 /
$$\frac{\left(1 - \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)}{2 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - \frac{1}{2} + \frac{1}{2 \sec{\left(2 x \right)}}$$
cot(x)
-1 - -----------
2
1 + cot (x)
$$-1 - \frac{\cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}$$
/ 2/x\ \ / 2/x pi\\
| cos |-| | | cos |- + --||
2/x pi\ | \2/ | | \2 4 /| / / pi\\
2 cos |- - --|*|1 - ------------|*|1 - ------------|*|1 + cos|x - --||
/ / pi\\ 2/x pi\ \2 2 / | 2/x pi\| | 2/x pi\| \ \ 2 //
|1 + cos|x - --|| *cos |- + --| | cos |- - --|| | cos |- - --||
1 cos(2*x) \ \ 2 // \2 4 / \ \2 2 // \ \2 4 //
- - + -------- - ------------------------------- + --------------------------------------------------------------------
2 2 2/x pi\ 2
cos |- - --|
\2 4 /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right) \left(- \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2} - \frac{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}} + \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2}$$
/ 2 \ / 2 \
2/x\ | cos (x) | | sin (x) |
sin |-|*|1 - --------------|*|1 - ---------|*(1 + sin(x))
/pi \ \2/ | 4/x pi\| | 4/x\|
sin|-- + 2*x| 2 2 | 4*sin |- + --|| | 4*sin |-||
1 \2 / (1 + sin(x)) *cos (x) \ \2 4 // \ \2//
- - + ------------- - --------------------- + ---------------------------------------------------------
2 2 4/x pi\ 2
4*sin |- + --|
\2 4 /
$$\frac{\left(1 - \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{\left(\sin{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{2} - \frac{1}{2}$$
/pi \ 4/x pi\
sin|-- + 2*x| 16*sin |- + --|
1 \2 / / 2/pi x\\ \2 4 /
- - + ------------- + |1 - 2*sin |-- + -||*sin(x) - -----------------------------
2 2 \ \2 2// 2
/ 4/x pi\\
| 4*sin |- + --||
| \2 4 /| 2
|1 + --------------| *cos (x)
| 2 |
\ cos (x) /
$$\left(1 - 2 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}\right) \sin{\left(x \right)} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{2} - \frac{1}{2} - \frac{16 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} \cos^{2}{\left(x \right)}}$$
1
-1 - ----------
2*csc(2*x)
$$-1 - \frac{1}{2 \csc{\left(2 x \right)}}$$
2/x pi\
4*tan |- + --|
1 - cos(2*x) / 2/x\\ \2 4 /
- ------------ + |1 - 2*cos |-||*sin(x) - -------------------
2 \ \2// 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$\left(1 - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(x \right)} - \frac{1 - \cos{\left(2 x \right)}}{2} - \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/x pi\
4*tan |- + --|
1 cos(2*x) sin(2*x) \2 4 /
- - + -------- - -------- - -------------------
2 2 2 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{\sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2} - \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ 2\
| / 2/x\\ |
| 2*|1 - tan |-|| |
| \ \4// | /x\
2*|1 - ----------------|*tan|-|
| 2 | \2/
2/x pi\ | / 2/x\\ |
2 4*tan |- + --| | |1 + tan |-|| |
1 1 - tan (x) \2 4 / \ \ \4// /
- - + --------------- - ------------------- + -------------------------------
2 / 2 \ 2 2/x\
2*\1 + tan (x)/ / 2/x pi\\ 1 + tan |-|
|1 + tan |- + --|| \2/
\ \2 4 //
$$\frac{1 - \tan^{2}{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{2 \left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{1}{2} - \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
/ 2/x\ \ / 2/x pi\ \
| csc |-| | | csc |- + --| |
/ 1 \ | \2/ | | \2 4 / |
2 |1 + ------|*|1 - ------------|*|1 - --------------|
/ 1 \ 2/x pi\ \ csc(x)/ | 2/pi x\| | 2/ x pi\|
|1 + ------| *csc |- + --| | csc |-- - -|| | csc |- - + --||
1 1 \ csc(x)/ \2 4 / \ \2 2// \ \ 2 4 //
- - + --------------- - -------------------------- + ----------------------------------------------------
2 /pi \ 2/ x pi\ 2/x\
2*csc|-- - 2*x| csc |- - + --| 2*csc |-|
\2 / \ 2 4 / \2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(- \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)}{2 \csc^{2}{\left(\frac{x}{2} \right)}} - \frac{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}} - \frac{1}{2} + \frac{1}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
2
1 - -------
2/x\ 2/x pi\
sec |-| 4*sec |- + --|
1 1 \2/ \2 4 /
- - + ---------- + ----------- - --------------------------------
2 2*sec(2*x) / pi\ 2
sec|x - --| / 2/x pi\\
\ 2 / | sec |- + --||
| \2 4 /| 2/x pi\
|1 + ------------| *sec |- - --|
| 2/x pi\| \2 4 /
| sec |- - --||
\ \2 4 //
$$\frac{1 - \frac{2}{\sec^{2}{\left(\frac{x}{2} \right)}}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{2} + \frac{1}{2 \sec{\left(2 x \right)}} - \frac{4 \sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}$$
sin(2*x) sin(x) sin(3*x)
1 + -------- - cos(x) - ------ - --------
2 4 4
-----------------------------------------
-1 + cos(x)
$$\frac{- \frac{\sin{\left(x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{2} - \frac{\sin{\left(3 x \right)}}{4} - \cos{\left(x \right)} + 1}{\cos{\left(x \right)} - 1}$$
2 2/ pi\
- sin (x) - sin |x + --| - cos(x)*sin(x)
\ 2 /
$$- \sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$
2/x pi\ 2/x\ / 2/x\\ / 2/x pi\\
4*cot |- + --| 4*tan |-| |1 - cot |-||*|1 - cot |- + --||
\2 4 / \2/ \ \2// \ \2 4 //
- ------------------- - -------------- + --------------------------------
2 2 / 2/x\\ / 2/x pi\\
/ 2/x pi\\ / 2/x\\ |1 + cot |-||*|1 + cot |- + --||
|1 + cot |- + --|| |1 + tan |-|| \ \2// \ \2 4 //
\ \2 4 // \ \2//
$$\frac{\left(1 - \cot^{2}{\left(\frac{x}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
2/x\ 2/x pi\ / 2/x\\ / 2/x pi\\
4*cot |-| 4*tan |- + --| |-1 + tan |-||*|-1 + tan |- + --||
\2/ \2 4 / \ \2// \ \2 4 //
- -------------- - ------------------- + ----------------------------------
2 2 / 2/x\\ / 2/x pi\\
/ 2/x\\ / 2/x pi\\ |1 + tan |-||*|1 + tan |- + --||
|1 + cot |-|| |1 + tan |- + --|| \ \2// \ \2 4 //
\ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
1 cos(2*x) (1 + sin(x)) 4/x\ 2/x\ / 1 \ / 1 \
- - + -------- - ------------- + 2*cos |-|*tan |-|*|1 - -------|*|1 - ------------|*(1 + sin(x))
2 2 2/x pi\ \4/ \4/ | 2/x\| | 2/x pi\|
tan |- + --| | tan |-|| | tan |- + --||
\2 4 / \ \2// \ \2 4 //
$$2 \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)} - \frac{\left(\sin{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2}$$
2/x pi\
4*cos |- - --|
1 cos(2*x) / 2/x\\ / pi\ \2 4 /
- - + -------- + |1 - 2*cos |-||*cos|x - --| - --------------------------------
2 2 \ \2// \ 2 / 2
/ 2/x pi\\
| cos |- - --||
| \2 4 /| 2/x pi\
|1 + ------------| *cos |- + --|
| 2/x pi\| \2 4 /
| cos |- + --||
\ \2 4 //
$$\left(1 - 2 \cos^{2}{\left(\frac{x}{2} \right)}\right) \cos{\left(x - \frac{\pi}{2} \right)} + \frac{\cos{\left(2 x \right)}}{2} - \frac{1}{2} - \frac{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
sin(2*x)
-1 - --------
2
$$- \frac{\sin{\left(2 x \right)}}{2} - 1$$
/ 1 \ / 1 \
|1 - -------|*|1 - ------------|
2/x\ | 2/x\| | 2/x pi\|
4*tan |-| | tan |-|| | tan |- + --||
\2/ 4 \ \2// \ \2 4 //
- -------------- - -------------------------------- + --------------------------------
2 2 / 1 \ / 1 \
/ 2/x\\ / 1 \ 2/x pi\ |1 + -------|*|1 + ------------|
|1 + tan |-|| |1 + ------------| *tan |- + --| | 2/x\| | 2/x pi\|
\ \2// | 2/x pi\| \2 4 / | tan |-|| | tan |- + --||
| tan |- + --|| \ \2// \ \2 4 //
\ \2 4 //
$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 - \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)} - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} - \frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
1 1 1
- ------- - ------------ - -------------
2 2/pi \ csc(x)*sec(x)
csc (x) csc |-- - x|
\2 /
$$- \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)}}$$
2
1 - ------------
2/pi x\ 2/ x pi\
csc |-- - -| 4*csc |- - + --|
1 1 \2 2/ \ 2 4 /
- - + --------------- + ---------------- - ----------------------------------
2 /pi \ csc(x) 2
2*csc|-- - 2*x| / 2/ x pi\\
\2 / | csc |- - + --||
| \ 2 4 /| 2/x pi\
|1 + --------------| *csc |- + --|
| 2/x pi\ | \2 4 /
| csc |- + --| |
\ \2 4 / /
$$\frac{1 - \frac{2}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}}{\csc{\left(x \right)}} - \frac{1}{2} + \frac{1}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{4 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}$$
1
-1 - ---------------
/ pi\
2*sec|2*x - --|
\ 2 /
$$-1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
2/x\ / 2/x\\ / 2/x pi\\
sin |-|*|1 - cot |-||*|1 - cot |- + --||*(1 + sin(x))
1 - cos(2*x) 2 2/x pi\ \2/ \ \2// \ \2 4 //
- ------------ - (1 + sin(x)) *cot |- + --| + -----------------------------------------------------
2 \2 4 / 2
$$- \frac{1 - \cos{\left(2 x \right)}}{2} + \frac{\left(1 - \cot^{2}{\left(\frac{x}{2} \right)}\right) \left(1 - \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right) \left(\sin{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{2} - \left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$
-(1 - cos(2*x))/2 - (1 + sin(x))^2*cot(x/2 + pi/4)^2 + sin(x/2)^2*(1 - cot(x/2)^2)*(1 - cot(x/2 + pi/4)^2)*(1 + sin(x))/2