Answer
$-\sqrt {3}$
Work Step by Step
Given: $\csc\theta =-2$
Recall: $\csc^{2}-\cot^{2}\theta=1$
$\implies \cot^{2}\theta=\csc^{2}\theta-1$
Or $\cot\theta=\pm\sqrt {\csc^{2}\theta-1}$
$\cos\theta$ is +ve and $\sin\theta$ is negative in the quadrant IV. Therefore, $\cot\theta=\frac{\cos\theta}{\sin\theta}$ is also negative in the quadrant IV.
$\implies \cot\theta=-\sqrt {\csc^{2}\theta-1}=-\sqrt {(-2)^{2}-1}$
$=-\sqrt {3}$
