[Math] Joint probability density function of cos(x) and sin(x)

Question

Assume x is uniformly distributed between 0 to 2pi.
How to get joint probability density function of cos(x) and sin(x)?

I know how to get the PDF for either cos(x) or sin(x), but have no idea about the joint PDF.

Answer 1

No random vector $U=(\cos(X),\sin(X))$ has a PDF since $P(U\in C)=1$ and the unit circle $C$ has zero two-dimensional Lebesgue measure. When $X$ is uniformly distributed on $(0,2\pi)$, all one can say is that $U$ is uniformly distributed on $C$.