Cosine of a uniform angle's mean given its sine

Let $X$ be uniformly distributed on $[0,\pi]$. Evaluate $\cos\big(\mathbb{E}[X\mid \sin X]\big)$.

$0$. Knowing $\sin X=s$ leaves two equally likely angles, $\arcsin s$ and $\pi-\arcsin s$, which are symmetric about $\tfrac\pi2$ (they merge when $s=1$). So $\mathbb{E}[X\mid\sin X]=\tfrac\pi2$ for every value, and $\cos\tfrac\pi2=0$.