Answer
The exact value of the provided expression is $\frac{4}{5}$.
Work Step by Step
Let us assume $\theta ={{\tan }^{-1}}\left( \frac{3}{4} \right)$. Then,
$\tan \theta =\frac{3}{4}$
As we know that $\tan \theta $ is positive, $\theta $ lies in the first quadrant.
Now, by using the Pythagorean theorem:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& {{r}^{2}}={{4}^{2}}+{{3}^{2}} \\
& r=\sqrt{16+9} \\
& r=5
\end{align}$
Then, the value of the given expression is
$\begin{align}
& \cos \left[ {{\tan }^{-1}}\frac{3}{4} \right]=\cos \theta \\
& =\frac{x}{r} \\
& =\frac{4}{5}
\end{align}$
