I found the following statement in Stoyanov's book on counterexamples (Section 7.2) quite interesting: Let $U_1$ and $U_2$ be independent and uniformly distributed on $(0,\pi)$. Then $\tan(U_1)$ and $\tan(U_1+U_2)$ are independent.
This is taken from https://projecteuclid.org/download/pdf_1/euclid.aoms/1177698885 and I am really confused by that: To my mind, they say in the last two sections that $U_1+U_2$ is also uniformly distributed and independent of $U_1$ (which would imply the statement) - this is clearly not true so that I must have misunderstood.
I would appreciate a clarification of this and I would like to know in what sense and why there actually is independence between $\tan(U_1)$ and $\tan(U_1+U_2)$.