cos(sin(... sin(x) ...))

[Mahfuz Sobhan]

If $x$ is small enough that $\sin x\to x$. An operator $\mbox{S}_n$ is defined such that $\mbox{S}_n\left(x\right)=\sin\left(\mbox{S}_{n-1}\left(x\right)\right)$ and $\mbox{S}_1\left(x\right)=\sin x$. For sufficiently large $n$, $\mbox{S}_n\left(x\right)\to \mbox{S}_{n-1}\left(x\right)$. Express $\cos\left(S_n\left(x\right)\right)$ to the nearest rational value.