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