Is there a way to simplify $\tan^{-1}(\cos x)$?

Question

I know how to rewrite trig functions of inverse trig functions, but this is pretty weird to me and I know it doesn't work the same way. I've tried seeing it as $\tan^{-1} x=f(\cos^{-1} x)$ for help, and writing it as the solution to a differential equation, as well as some simple substitutions, however nothing has worked. What are the simplest ways of rewriting this and are there any that don't involve infinite series?

Answer 1

There is no real simplification possible for $\arctan(\cos(x))$. Well, you could write it as $$ \arcsin\left(\frac{\cos(x)}{\sqrt{1+\cos^2(x)}}\right)$$