# quick question about reciprocal identity

By definition, $\csc x = 1 / \sin x$, so $\csc x$ is undefined for $x=\pi k$, $k\in\mathbb{Z}$. Therefore, $\csc x\ne0$ for all x and $\sin x = 1 / \csc x$ for $x\ne\pi k, k\in\mathbb{Z}$.