Let $R_0$ be a real number and $n$ an arbitrary integer
$$ \int\limits_{0}^{2 \pi} \cos( n \arctan( R_0 \sin(\vartheta))) \cos( n \arctan( R_0 \cos(\vartheta))) \; d \vartheta.$$
I have tried using $2 \cos(a) \cos(b) = \cos(a+b) +\cos(a-b)$, and also some substitutions. I would like to know how this integral behave with respect to $R_0$.
E.g. you can get something like
$$\int\limits_{0}^{2 \pi} \cos( n \vartheta) \cos( n \arctan( \sqrt{ R_0^2 -\tan^2(\vartheta)})) \frac{1+\tan^2(\vartheta)}{\sqrt{ R_0^2 -\tan^2(\vartheta))}} d\; \vartheta.$$
I expect Bessel integrals to turn up, but I am not sure about this.
This question is related, but less specific: An integral involving trigonometric functions and its inverse