Let $\operatorname{cosec}^{-1} x=0$, then $x=\operatorname{cosec} \theta$ and $\theta \in\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]$
$\therefore \quad \cot \left(\operatorname{cosec}^{-1} x\right)$
$-\sqrt{\operatorname{cosec}^2 \theta-1} \text { if }-\frac{\pi}{2} \leq \theta<0 \\$
$\sqrt{\operatorname{cosec}^2 \theta-1} \text { if } 0<\theta \leq \frac{\pi}{2}$
$-\sqrt{x^2-1} \text { if } x \leq-1 \\$
$\sqrt{x^2-1} \text { if } x \geq 1$
