For what integer $m$ and $n$ is $\tan(m \, \arctan(x) + n \, \arctan(y))$ a rational function of $x$ and $y$?

Question

For what values of $n,m \in N$ is $$\tan(m \, \arctan(x) + n \, \arctan(y))$$ rational?

I.e., what $n$ and $m$ values would allow me to simplify the expression above to a rational expression in $F[x,y]$, where $F$ is a ring or a field.