Mister Exam
How do you cos^2a/(1-cos^2a) in partial fractions?
The solution
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2
cos (a)
-----------
2
1 - cos (a)
$$\frac{\cos^{2}{\left(a \right)}}{1 - \cos^{2}{\left(a \right)}}$$
cos(a)^2/(1 - cos(a)^2)
Fraction decomposition
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-1 + 1/(2*(1 + cos(a))) - 1/(2*(-1 + cos(a)))
$$-1 + \frac{1}{2 \left(\cos{\left(a \right)} + 1\right)} - \frac{1}{2 \left(\cos{\left(a \right)} - 1\right)}$$
1 1
-1 + -------------- - ---------------
2*(1 + cos(a)) 2*(-1 + cos(a))
General simplification
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1 ------- 2 tan (a)
$$\frac{1}{\tan^{2}{\left(a \right)}}$$
tan(a)^(-2)
Common denominator
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1
-1 - ------------
2
-1 + cos (a)
$$-1 - \frac{1}{\cos^{2}{\left(a \right)} - 1}$$
-1 - 1/(-1 + cos(a)^2)
Combinatorics
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2
-cos (a)
--------------------------
(1 + cos(a))*(-1 + cos(a))
$$- \frac{\cos^{2}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} + 1\right)}$$
-cos(a)^2/((1 + cos(a))*(-1 + cos(a)))
Powers
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2
/ I*a -I*a\
|e e |
|---- + -----|
\ 2 2 /
-------------------
2
/ I*a -I*a\
|e e |
1 - |---- + -----|
\ 2 2 /
$$\frac{\left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}}{1 - \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2}}$$
(exp(i*a)/2 + exp(-i*a)/2)^2/(1 - (exp(i*a)/2 + exp(-i*a)/2)^2)
Trigonometric part
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2
/ 2/a pi\\
| sec |- - --||
| \2 2 /| 2/ pi\
|-1 + ------------| *sec |a - --|
| 2/a\ | \ 2 /
| sec |-| |
\ \2/ /
---------------------------------
4/a pi\
sec |- - --|
\2 2 /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
2 2
/ /a\\ / /a\\
|1 + tan|-|| *|-1 + tan|-|| *(1 - cos(a))
\ \2// \ \2//
-----------------------------------------
4/a\
4*(1 + cos(a))*tan |-|
\2/
$$\frac{\left(1 - \cos{\left(a \right)}\right) \left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \left(\cos{\left(a \right)} + 1\right) \tan^{4}{\left(\frac{a}{2} \right)}}$$
4/a\
4*sin |-|
\2/
----------------------
2 2
(-1 + cos(a)) *tan (a)
$$\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\left(\cos{\left(a \right)} - 1\right)^{2} \tan^{2}{\left(a \right)}}$$
2 cot (a)
$$\cot^{2}{\left(a \right)}$$
2
/ 2/a\ \
| sec |-| |
| \2/ | 2/ pi\
|1 - ------------| *sec |a - --|
| 2/a pi\| \ 2 /
| sec |- - --||
\ \2 2 //
--------------------------------
4/a\
sec |-|
\2/
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{4}{\left(\frac{a}{2} \right)}}$$
1 --------------------- / 1 \ 2 |1 - -------|*sec (a) | 2 | \ sec (a)/
$$\frac{1}{\left(1 - \frac{1}{\sec^{2}{\left(a \right)}}\right) \sec^{2}{\left(a \right)}}$$
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/| 2
|1 - ------------| *csc (a)
| 2/a\ |
| csc |-| |
\ \2/ /
---------------------------
4/pi a\
csc |-- - -|
\2 2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc^{2}{\left(a \right)}}{\csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
2
/ 2/a\ \
| cos |-| |
| \2/ | 4/a pi\
|-1 + ------------| *cos |- - --|
| 2/a pi\| \2 2 /
| cos |- - --||
\ \2 2 //
---------------------------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2} \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}$$
2
/ 2 \
| sin (a) | 4/a\
|-1 + ---------| *sin |-|
| 4/a\| \2/
| 4*sin |-||
\ \2//
-------------------------
2
sin (a)
$$\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}}$$
2 2
/ /a\\ / /a\\
|1 + tan|-|| *|-1 + tan|-|| *(1 + cos(a))
\ \2// \ \2//
-----------------------------------------
4*(1 - cos(a))
$$\frac{\left(\cos{\left(a \right)} + 1\right) \left(\tan{\left(\frac{a}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \left(1 - \cos{\left(a \right)}\right)}$$
2/ pi\
sec |a - --|
\ 2 /
------------
2
sec (a)
$$\frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}$$
2
/ 2/a\\
|-1 + cot |-||
\ \2//
------------------------------------
/ 2\
| / 2/a\\ |
2 | |-1 + cot |-|| |
/ 2/a\\ | \ \2// |
|1 + cot |-|| *|1 - ---------------|
\ \2// | 2|
| / 2/a\\ |
| |1 + cot |-|| |
\ \ \2// /
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(- \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 2 4
/ 2/a\\ / 1 \ / 2/a\\
|1 + cot |-|| *|1 - -------| *|-1 + cot |-||
\ \2// | 2/a\| \ \4//
| cot |-||
\ \2//
---------------------------------------------
4
/ 2/a\\ 2/a\
4*|1 + cot |-|| *cot |-|
\ \4// \2/
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4} \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{a}{2} \right)}}$$
1 ------------------------------- / 1 \ 2/pi \ |1 - ------------|*csc |-- - a| | 2/pi \| \2 / | csc |-- - a|| \ \2 //
$$\frac{1}{\left(1 - \frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
2
/ 2/a\\ 4/a\
|1 - tan |-|| *cos |-|
\ \2// \2/
----------------------
2
sin (a)
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \cos^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}}$$
1 1
-1 + --------- + ---------
2/a\ 2/a\
4*cos |-| 4*sin |-|
\2/ \2/
$$-1 + \frac{1}{4 \cos^{2}{\left(\frac{a}{2} \right)}} + \frac{1}{4 \sin^{2}{\left(\frac{a}{2} \right)}}$$
2
sin (2*a)
---------
4
4*sin (a)
$$\frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}$$
2 2
/ 2/a\\ / 2/a\\ 4/a\
4*|1 + cot |-|| *|-1 + cot |-|| *cot |-|
\ \2// \ \2// \4/
----------------------------------------
4
/ 2/a\\ 2/a\
|1 + cot |-|| *cot |-|
\ \4// \2/
$$\frac{4 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\\ 4/a\
|-1 + cot |-|| *sin |-|
\ \2// \2/
-----------------------
2
sin (a)
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}}$$
2/ pi\
sin |a + --|
\ 2 /
----------------
2/ pi\
1 - sin |a + --|
\ 2 /
$$\frac{\sin^{2}{\left(a + \frac{\pi}{2} \right)}}{1 - \sin^{2}{\left(a + \frac{\pi}{2} \right)}}$$
2
/ 4/a\\
| 4*sin |-||
| \2/| 4/pi a\
|1 - ---------| *sin |-- + -|
| 2 | \2 2/
\ sin (a) /
-----------------------------
2
sin (a)
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(a \right)}}$$
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /| 4/a\
|1 - ------------| *cos |-|
| 2/a\ | \2/
| cos |-| |
\ \2/ /
---------------------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{4}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}$$
2 4
/ 2/a\\ / 2/a\\ 8/a\ 2/a\
|1 - tan |-|| *|-1 + cot |-|| *sin |-|*tan |-|
\ \2// \ \4// \4/ \2/
----------------------------------------------
4/a\
4*sin |-|
\2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}$$
2
csc (a)
------------
2/pi \
csc |-- - a|
\2 /
$$\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}$$
1 ------- 2 tan (a)
$$\frac{1}{\tan^{2}{\left(a \right)}}$$
2
cos (a)
------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}$$
4/a\
sin |-| 2
1 \2/ sin (a)
- - + ------- + ----------
2 2 4/a\
sin (a) 16*sin |-|
\2/
$$\frac{\sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} - \frac{1}{2} + \frac{\sin^{2}{\left(a \right)}}{16 \sin^{4}{\left(\frac{a}{2} \right)}}$$
2 4
/ 2/a\\ / 2/a\\ 8/a\
|1 - tan |-|| *|1 - tan |-|| *cos |-|
\ \2// \ \4// \4/
-------------------------------------
4/a\ 2/a\
4*cos |-|*tan |-|
\2/ \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \cos^{8}{\left(\frac{a}{4} \right)}}{4 \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}$$
2 2 4
/ 2/a\\ / 2/a\\ / 2/a\\
|1 + tan |-|| *|1 - tan |-|| *|1 - tan |-||
\ \2// \ \2// \ \4//
--------------------------------------------
4
/ 2/a\\ 2/a\
4*|1 + tan |-|| *tan |-|
\ \4// \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\\
|1 - tan |-||
\ \2//
-----------------------------------
/ 2\
| / 2/a\\ |
2 | |1 - tan |-|| |
/ 2/a\\ | \ \2// |
|1 + tan |-|| *|1 - --------------|
\ \2// | 2|
| / 2/a\\ |
| |1 + tan |-|| |
\ \ \2// /
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 2
/ 2/a\\ / 1 \ 4/a\
4*|1 + tan |-|| *|-1 + -------| *tan |-|
\ \2// | 2/a\| \4/
| tan |-||
\ \2//
----------------------------------------
4
/ 2/a\\ 2/a\
|1 + tan |-|| *tan |-|
\ \4// \2/
$$\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{a}{2} \right)}}$$
2 / 1 \ 2 4/a\ |-1 + -------| *csc (a)*sin |-| | 2/a\| \2/ | tan |-|| \ \2//
$$\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)} \csc^{2}{\left(a \right)}$$
2
/ 2/a\ \
| csc |-| |
| \2/ | 2
|-1 + ------------| *csc (a)
| 2/pi a\|
| csc |-- - -||
\ \2 2//
----------------------------
4/a\
csc |-|
\2/
$$\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2} \csc^{2}{\left(a \right)}}{\csc^{4}{\left(\frac{a}{2} \right)}}$$
(-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2*csc(a)^2/csc(a/2)^4