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