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