Conditional expectation of $\cos (2\pi X)$ on $\sin (2\pi X)$? Are they mean independent?

Question

Let $X$ be a random variable that follows uniform distribution in $[0,1]$.

what is the expectation of $\cos (2\pi X)$ conditional on $\sin (2\pi X)$?

What I have thought about is this is not as simple as given $X$, since $\sin (2\pi X)$ is periodic. Given $\sin (2\pi X)$ equals some value actually gives a sequence of plausible values for $X$.

Answer 1

Since $\cos(2\pi X) = \text{sign}(\cos(2\pi X))|\cos(2\pi X)| = \text{sign}(\cos(2\pi X)) \sqrt{1-\sin^2(2\pi X)}$, we have $$E(\cos(2\pi X)|\sin(2\pi X)) = \sqrt{1-\sin^2(2\pi X)} E[\text{sign}(\cos(2\pi X))|\sin(2\pi X)]$$

Note that $E[\text{sign}(\cos(2\pi X))|\sin(2\pi X)] = E[1_{\cos(2\pi X)>0}|\sin(2\pi X)]- E[1_{\cos(2\pi X)\leq 0}|\sin(2\pi X)]$ and for reasons of symmetry we have $E[1_{\cos(2\pi X)>0}|\sin(2\pi X)]= E[1_{\cos(2\pi X)\leq 0}|\sin(2\pi X)]$. Thus $$E(\cos(2\pi X)|\sin(2\pi X)) = 0$$