Mister Exam
Least common denominator atan(tan(f)*sqrt(a^2-1)/a)/sqrt(a^2-1)
The solution
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/ ________\
| / 2 |
|tan(f)*\/ a - 1 |
atan|------------------|
\ a /
------------------------
________
/ 2
\/ a - 1
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \tan{\left(f \right)}}{a} \right)}}{\sqrt{a^{2} - 1}}$$
atan((tan(f)*sqrt(a^2 - 1))/a)/sqrt(a^2 - 1)
Trigonometric part
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/ _________ \
| / 2 |
|\/ -1 + a *sec(f)|
atan|-------------------|
| / pi\ |
| a*sec|f - --| |
\ \ 2 / /
-------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \sec{\left(f \right)}}{a \sec{\left(f - \frac{\pi}{2} \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________ \
| / 2 /pi \|
|\/ -1 + a *csc|-- - f||
| \2 /|
atan|------------------------|
\ a*csc(f) /
------------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \csc{\left(- f + \frac{\pi}{2} \right)}}{a \csc{\left(f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________ \
| / 2 |
|\/ -1 + a *sin(f)|
atan|-------------------|
\ a*cos(f) /
-------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \sin{\left(f \right)}}{a \cos{\left(f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________\
| / 2 |
|\/ -1 + a |
atan|------------|
\ a*cot(f) /
------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1}}{a \cot{\left(f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________ \
| / 2 |
|\/ -1 + a *sec(f)|
atan|-------------------|
\ a*csc(f) /
-------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \sec{\left(f \right)}}{a \csc{\left(f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________ \
| / 2 / pi\|
|\/ -1 + a *cos|f - --||
| \ 2 /|
atan|------------------------|
\ a*cos(f) /
------------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \cos{\left(f - \frac{\pi}{2} \right)}}{a \cos{\left(f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
/ _________ \
| / 2 2 |
|2*\/ -1 + a *sin (f)|
atan|----------------------|
\ a*sin(2*f) /
----------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{2 \sqrt{a^{2} - 1} \sin^{2}{\left(f \right)}}{a \sin{\left(2 f \right)}} \right)}}{\sqrt{a^{2} - 1}}$$
atan(2*sqrt(-1 + a^2)*sin(f)^2/(a*sin(2*f)))/sqrt(-1 + a^2)
Combinatorics
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/ _________ \
| / 2 |
|\/ -1 + a *tan(f)|
atan|-------------------|
\ a /
-------------------------
__________________
\/ (1 + a)*(-1 + a)
$$\frac{\operatorname{atan}{\left(\frac{\sqrt{a^{2} - 1} \tan{\left(f \right)}}{a} \right)}}{\sqrt{\left(a - 1\right) \left(a + 1\right)}}$$
atan(sqrt(-1 + a^2)*tan(f)/a)/sqrt((1 + a)*(-1 + a))
Numerical answer
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(-1.0 + a^2)^(-0.5)*atan((tan(f)*sqrt(a^2 - 1))/a)
(-1.0 + a^2)^(-0.5)*atan((tan(f)*sqrt(a^2 - 1))/a)
Powers
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/ _________ \
| / 2 / I*f -I*f\|
|I*\/ -1 + a *\- e + e /|
atan|-------------------------------|
| / I*f -I*f\ |
\ a*\e + e / /
-------------------------------------
_________
/ 2
\/ -1 + a
$$\frac{\operatorname{atan}{\left(\frac{i \sqrt{a^{2} - 1} \left(- e^{i f} + e^{- i f}\right)}{a \left(e^{i f} + e^{- i f}\right)} \right)}}{\sqrt{a^{2} - 1}}$$
atan(i*sqrt(-1 + a^2)*(-exp(i*f) + exp(-i*f))/(a*(exp(i*f) + exp(-i*f))))/sqrt(-1 + a^2)