Graph of the Inverse Cosecant Function

$y=\text{cosec}^{-1}x$ has domain $\mathbb{R}\setminus(-1,1)$ and range $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]\setminus\{0\}$ — that is, $\text{cosec}^{-1}:\mathbb{R}\setminus(-1,1)\to\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]\setminus\{0\}$.

Fig. 4.21 shows the restricted cosecant curve on its principal domain, and Fig. 4.22 its reflection in $y=x$: two flattening branches, one for $x\ge1$ approaching the horizontal asymptote $y=0^+$ near $x=\tfrac{\pi}2$'s side and one for $x\le-1$ symmetric below, with a genuine gap in the graph (matching the excluded $y=0$ in the range) directly abo …