Answer
The exact value of the provided expression is $-\frac{3}{4}$.
Work Step by Step
Let us assume $\theta ={{\sin }^{-1}}\left( -\frac{3}{5} \right)$. Then,
$\sin \theta =-\frac{3}{5}$
As we know that $\sin \theta $ is negative, $\theta $ lies in the fourth quadrant.
Now, by using the Pythagorean theorem:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& {{5}^{2}}={{x}^{2}}+{{\left( -3 \right)}^{2}} \\
& x=\sqrt{25-9} \\
& x=4
\end{align}$
Then, the value of the given expression is
$\begin{align}
& \tan \left[ {{\sin }^{-1}}\left( -\frac{3}{5} \right) \right]=\tan \theta \\
& =\frac{y}{x} \\
& =-\frac{3}{4}
\end{align}$
