Maybe rewriting your equation as
$$ x \tan f = y$$ does help?
Given the fact that the first hint did not help. Here, is the second hint: you can rewrite your equation as
$$ F(x,y,f) = x \tan f - y =0.$$
Can you then use the implicit function theorem to learn something about $\partial_x f$ and $\partial_y f$?
I just did see that you have changed your question and thus the points with $f=\pi/2 + n\pi$ are not excluded any more. In this case you should rewrite your equation as (check the special points at $f=\pi/2 + n\pi$ separately)
$$F(x,y,f) = x \sin f - y \cos f =0.$$
and then apply the implicit function theorem.