Answer
$-\displaystyle \frac{1}{\sqrt{3}}$
Work Step by Step
$y=\sin^{-1}x$ is the number in $[-\pi/2, \pi/2]$ for which $\sin y=x.$
Sine is an odd function, and $\displaystyle \sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2} \quad $(QI)
which leads to $\displaystyle \sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$, so
$\displaystyle \cot(\sin^{-1}(-\frac{\sqrt{3}}{2}))=\cot(-\frac{\pi}{3})=$
We note that cot is also an odd function; thus:
$=-\displaystyle \cot\frac{\pi}{3}$
$=-\displaystyle \frac{1}{\sqrt{3}}$
