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