Answer
The exact value of the provided expression is $-\frac{\sqrt{3}}{3}$.
Work Step by Step
Let us assume $\theta ={{\sin }^{-1}}\left( -\frac{1}{2} \right)$. Then
$\sin \theta =-\frac{1}{2}$
We know that for the sine function, the interval for the angle is $\left[ -\frac{\pi }{2},\frac{\pi }{2} \right]$.
Therefore, the only angle that satisfies $\sin \theta =-\frac{1}{2}$ is $\left( -\frac{\pi }{6} \right)$.
Hence, $\theta =\left( -\frac{\pi }{6} \right)$ and,
The exact value of the given expression is
$\begin{align}
& \tan \left[ {{\sin }^{-1}}\left( -\frac{1}{2} \right) \right]=\tan \left( \theta \right) \\
& =\tan \left( -\frac{\pi }{6} \right) \\
& =-\frac{\sqrt{3}}{3}
\end{align}$
