Let $Z$ and $\theta \sim \text{Unif}[0, 2\pi)$ be two independent random variables, in which $Z$ has the following pdf: $$ f_Z(z) = \begin{cases} 0, & z \leq 0 \\ ze^{-z^2/2}, & z > 0\end{cases}$$
Show that $A = Z\cos(\theta)$ and $B = Z\sin(\theta)$ are also independent.
My attempt was to use $f_{A,B}(a, b) = \frac{d}{db} F_{B|A}(b | a) f_A(a)$ and show that $f_{A, B}(a, b) = f_A(a) f_B(b)$, but nothing came out.
I think knowing that $Z = \sqrt{A^2 + B^2}$ would somehow help, but I can't see where to use that.
If $X$ and $Y$ are independent standard normal random variables, then using polar coordinates you can write (for any arbitrary bounded measurable function $g$) $$ E[g(X,Y,\sqrt{X^2+Y^2})] =\int_0^{2\pi} \int_0^\infty g(r\cos\theta,r\sin\theta,r) r {e^{-r^2/2}} dr{1\over 2\pi} d\theta. $$ This shows that the joint distribution of $(A,B,Z)$ is the same as that of $(X,Y,\sqrt{X^2+Y^2})$. In particular, $A$ and $B$ are independent.