# 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?

$$\arctan(a \tan(x)) = \arcsin\left(\frac{a \sin(x)}{\sqrt{\cos(x)^2 + a^2 \sin(x)^2}}\right)$$