Answer
The exact value of the provided expression is $-\frac{3}{4}$.
Work Step by Step
Let us assume $\theta ={{\cos }^{-1}}\left( -\frac{4}{5} \right)$. Then,
$\cos \theta =-\frac{4}{5}$
As we know $\cos \theta $ is negative, $\theta $ lies in the second quadrant.
Now, by using the Pythagorean theorem:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& {{5}^{2}}={{\left( -4 \right)}^{2}}+{{y}^{2}} \\
& y=\sqrt{25-16} \\
& y=3
\end{align}$
Then, the value of the given expression is
$\begin{align}
& \tan \left[ {{\cos }^{-1}}\left( -\frac{4}{5} \right) \right]=\tan \theta \\
& =\frac{y}{x} \\
& =-\frac{3}{4}
\end{align}$
