### Reduction to a rational expression

I have the the following expression:

$ tan(m \cdot actan(x) + n \cdot arctan(y))$

For what values of $n,m \in \mathbb{N}$ is this expression reducible to a rational expression in ~~$R[x,y], ~~$R[x,y]$, where ~~R ~~$R$ is generic ring$?

Is there some rule $lhs => rhs $ that is needed in order to get a rational expression form the expression ~~above? ~~above?