Cosine of a conditional mean

Let $X \sim \text{Uniform}([0, \pi])$ . Find $\cos\!\big(\mathbb{E}[X \mid \sin X]\big)$ .

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Given $\sin X = s$ , which values of $X$ in $[0,\pi]$ are possible, and how likely is each?
The two solutions are symmetric about $\pi/2$ , so the conditional mean is $\pi/2$ .

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