Inverse Cosecant Function

Consider $\text{cosec}$ restricted to $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)-\{0\}\to\mathbb{R}-(-1,1)$. Graphically this restriction is one-one and onto $\mathbb{R}-(-1,1)$, so its inverse exists, called the inverse cosecant function, denoted $\text{cosec}^{-1}:\mathbb{R}-(-1,1)\to\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)-\{0\}$. For $x\in\mathbb{R}-(-1,1)$ and $\theta\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)-\{0\}$, we write $\text{cosec}^{-1}x=\theta$ if $\text{cosec}\,\theta=x$.

Ex. $\text{cosec}\,\dfrac{\pi}{6}=2$, where $2\in\mathbb{R}-(-1,1)$ and $\dfrac{\pi}{6}\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)-\{0\}$, so $\text{cosec}^{-1}(2)=\dfrac{\pi}{6}$: the principal value of $\text{cosec}^{-1}2$ is $\dfrac{\pi}{6}$. …