Answer
The exact value of the provided expression is $-\frac{\sqrt{3}}{3}$.
Work Step by Step
Let us assume $\theta ={{\cos }^{-1}}\left( -\frac{\sqrt{3}}{2} \right)$. Then,
$\cos \theta =-\frac{\sqrt{3}}{2}$
We know that for the cosine function, the interval for the angle is $\left[ 0,\pi \right]$.
Thus, the only angle that satisfies $\cos \theta =-\frac{\sqrt{3}}{2}$ is $\left( \frac{5\pi }{6} \right)$.
Hence, $\theta =\left( \frac{5\pi }{6} \right)$ and the exact value of the given expression is
$\begin{align}
& \tan \left[ {{\cos }^{-1}}\left( -\frac{\sqrt{3}}{2} \right) \right]=\tan \left( \theta \right) \\
& =\tan \left( \frac{5\pi }{6} \right) \\
& =-\frac{\sqrt{3}}{3}
\end{align}$
