Suppose $\theta \sim \text{Uniform}[0,2\pi]$, and $X=\cos(\theta)\,,\, Y=\sin(\theta)$.
Let two events be defined by $A=\{X\geq 0\}\,,\, B=\{Y\geq 0\}$.
I want to find the correlation coefficient between $X$ and $Y$ conditioned on $A \cup B$. But I don't know how to proceed. Any help?
Since $A\cup B$ is equivalent to $\theta \in \left[0, \pi\right] \cup \left[\frac{3\pi}2, 2\pi\right]$, $$\mathbb E\left[X\Big | A \cup B\right] = \frac{\frac1{2\pi}\displaystyle\int_{\left[0, \pi\right] \cup \left[\frac{3\pi}2, 2\pi\right]} X(\theta) \mathrm d \theta}{\frac1{2\pi}\displaystyle\int_{\left[0, \pi\right] \cup \left[\frac{3\pi}2, 2\pi\right]} \mathrm d \theta} = \frac{\displaystyle\int_0^\pi \cos(\theta) \mathrm d \theta + \int_{\frac{3\pi}2}^{2\pi} \cos(\theta) \mathrm d \theta}{\displaystyle\int_0^\pi \mathrm d \theta + \int_{\frac{3\pi}2}^{2\pi} \mathrm d \theta} = \frac{0 + 1}{\pi + \frac\pi 2} = \frac2{3\pi}$$
Doing the same thing for $Y$ you have $$\mathbb E\left[X\Big | A \cup B\right] = \frac2{3\pi}$$
Now for $XY$ $$\mathbb E\left[XY\Big | A \cup B\right] = \frac{\frac1{2\pi}\displaystyle\int_{\left[0, \pi\right] \cup \left[\frac{3\pi}2, 2\pi\right]} X(\theta) Y(\theta)\mathrm d \theta}{\frac1{2\pi}\displaystyle\int_{\left[0, \pi\right] \cup \left[\frac{3\pi}2, 2\pi\right]} \mathrm d \theta} = \frac{\displaystyle\int_0^\pi \cos(\theta)\sin(\theta) \mathrm d \theta + \int_{\frac{3\pi}2}^{2\pi} \cos(\theta)\sin(\theta) \mathrm d \theta}{\displaystyle\int_0^\pi \mathrm d \theta + \int_{\frac{3\pi}2}^{2\pi} \mathrm d \theta} = \frac{\displaystyle\int_0^\pi \frac12\sin(2\theta) \mathrm d \theta + \int_{\frac{3\pi}2}^{2\pi} \frac12\sin(2\theta) \mathrm d \theta}{\pi + \frac\pi 2} = \frac{\frac12 }{\frac{3\pi} 2} = \frac1{3\pi}$$