How do I find the probability distribution of a dependent variable given the probability distributions of its independent variables?

I have two probability distribution functions, $F$ and $G$, describing quantities $x$ and $y$ respectively.

Both $x$ and $y$ are required to find quantity $z$ – let’s say $z = x^2(1-cos(y))$.

$x$ is defined over a range which we can call [x₁, x₂], while y is defined over [0,2π]. Both are continuous.

Given that I have $F$ and $G$, how can I find the probability distribution for $z$?

Thanks in advance!