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