The Cosecant Function and the Inverse Cosecant Function

$\text{cosec}\,x=\dfrac1{\sin x}$ is, like sine, an ODD function with period $2\pi$ — its values repeat over every interval of length $2\pi$. Since $\text{cosec}\,x$ is undefined wherever $\sin x=0$, its domain is $\mathbb{R}\setminus\{n\pi:n\in\mathbb{Z}\}$. Because $-1\le\sin x\le1$ (and $\sin x\ne0$ in the domain), $\text{cosec}\,x$ never takes a value strictly …