Calculating $\mathbb{E}[2\sin (\pi Z)|\cos (\pi Z)]$ when $Z$ is uniform on $[0,2]$ I am trying to calculate the following conditional expectation.
Let Z be a uniformly distributed random variable on the closed interval $[0, 2]$.
Define $X = \cos(\pi Z)$ and $Y = 2\sin(\pi Z)$.
Calculate $\mathbb{E}[Y|X] = \mathbb{E}[Y|\sigma(X)]$.
I have tried multiple approaches, but don't know how to proceed since I am only familiar with the standard techniques for calculating the conditional expectation, i.e. when Y is independent of X, Y is measurable with respect to $\sigma(X)$ or if a joint density $f(x,y)$ exists. The first two parts don't apply here and since X and Y aren't independent I don't know how to calculate the joint density.
Any help is greatly appreciated. Thank you very much!

This is a simplified case from:
https://stats.stackexchange.com/questions/603110/how-to-calculate-px-in-a-sinx-y
Here: $P[Z\in A|X=x]=\frac{1}{2}(I_A(arccos(x)/\pi)+I_A((2\pi-arccos(x))/\pi))$
Therefore: $E[X|Y]=E[Y|X]=0$
