Mister Exam
How do you (2tgx)/(1+tg^2x) in partial fractions?
The solution
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2*tan(x)
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2
1 + tan (x)
$$\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
(2*tan(x))/(1 + tan(x)^2)
Fraction decomposition
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2*tan(x)/(1 + tan(x)^2)
$$\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
2*tan(x)
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2
1 + tan (x)
Powers
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/ I*x -I*x\
2*I*\- e + e /
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/ 2\
| / I*x -I*x\ |
| \- e + e / | / I*x -I*x\
|1 - -----------------|*\e + e /
| 2 |
| / I*x -I*x\ |
\ \e + e / /
$$\frac{2 i \left(- e^{i x} + e^{- i x}\right)}{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right)}$$
2*i*(-exp(i*x) + exp(-i*x))/((1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x)))
Trigonometric part
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2*sec(x) -------------------- / 2 \ | sec (x)| |1 + -------|*csc(x) | 2 | \ csc (x)/
$$\frac{2 \sec{\left(x \right)}}{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$
1 ------------- / pi\ sec|2*x - --| \ 2 /
$$\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}$$
/ pi\
2*cos|x - --|
\ 2 /
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/ 2/ pi\\
| cos |x - --||
| \ 2 /|
|1 + ------------|*cos(x)
| 2 |
\ cos (x) /
$$\frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}$$
2 -------------------- / 1 \ |1 + -------|*cot(x) | 2 | \ cot (x)/
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}$$
sin(2*x)
$$\sin{\left(2 x \right)}$$
2*sin(x) -------------------- / 2 \ | sin (x)| |1 + -------|*cos(x) | 2 | \ cos (x)/
$$\frac{2 \sin{\left(x \right)}}{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)}}$$
1 -------- csc(2*x)
$$\frac{1}{\csc{\left(2 x \right)}}$$
/pi \
2*csc|-- - x|
\2 /
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/ 2/pi \\
| csc |-- - x||
| \2 /|
|1 + ------------|*csc(x)
| 2 |
\ csc (x) /
$$\frac{2 \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$
2*sec(x) ------------------------------ / 2 \ | sec (x) | / pi\ |1 + ------------|*sec|x - --| | 2/ pi\| \ 2 / | sec |x - --|| \ \ 2 //
$$\frac{2 \sec{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
2
4*sin (x)
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/ 4 \
| 4*sin (x)|
|1 + ---------|*sin(2*x)
| 2 |
\ sin (2*x)/
$$\frac{4 \sin^{2}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}$$
/ pi\ cos|2*x - --| \ 2 /
$$\cos{\left(2 x - \frac{\pi}{2} \right)}$$
2*cot(x)
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2
1 + cot (x)
$$\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}$$
2*cot(x)/(1 + cot(x)^2)