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

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?

• Where does this problem come from? It might be possible to simplify it at a step before this one. May 1, 2019 at 15:36
• It isn't from anything, but I stumbled across it trying to prove that arctan equals arcsin/arccos leads to a contradiction. May 1, 2019 at 15:41
• I don't think there is an easier form to write this without using a Taylor series or complex exponentials. May 1, 2019 at 15:53

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