Q: Can one simplify $\arctan(a\tan(x))$? We know that $\arctan(\tan(x))=x$ when $x$ lies between $-\pi/2$ and $+\pi/2$; but do you know a way to transform the expression $\arctan(a\tan(x))$, where $a$ is a real number between $0$ and $1$?
I thought $a$ could be transformed with trigonometric functions, such as $a=\sin(\alpha)\cos(x)$, but $\arctan(\sin(\alpha)\sin(x))$ does not remind me anything.
Maybe there is no further possible transformation?

A: Well, you can convert the arctan to other inverse-trig functions, such as arcsin:
$$ \arctan(a \tan(x)) = \arcsin\left(\frac{a \sin(x)}{\sqrt{\cos(x)^2 + a^2 \sin(x)^2}}\right) $$
