My book says "prove that if $z\in\mathbb C$ and $\lvert\sin z\rvert\le 1$, then $z\in\mathbb R$."
But I think this can't be true as
and so if $\lvert\sin z\rvert\le 1$, then, $\sinh^2y\le1-\sin^2x=\cos^2x$.
Clearly we can find some $y\neq 0$, such that $\sinh^2y\le\cos^2x$ for some $x$.
Thus I want to know if something went wrong in my explanations?