Answer
The exact value of $\cos \left( {{\tan }^{-1}}\frac{3}{4} \right)$ is $\frac{4}{5}$.
Work Step by Step
Let $\theta ={{\tan }^{-1}}\frac{3}{4}$ represent the angle in $\left( -\frac{\pi }{2},\frac{\pi }{2} \right)$.
$\begin{align}
& \theta ={{\tan }^{-1}}\frac{3}{4} \\
& \tan \theta =\frac{3}{4}
\end{align}$
As the value of $\tan \theta $ is positive, thus $\theta $ lies in the first quadrant.
Hence, the measures of the two sides of the right triangle are $3$ and $4$. The hypotenuse of the triangle is found by applying the Pythagorian Theorem,
$\begin{align}
& r=\sqrt{{{3}^{2}}+{{4}^{2}}} \\
& =\sqrt{25} \\
& =5
\end{align}$
From above sketch
$\begin{align}
& \theta ={{\tan }^{-1}}\frac{3}{4} \\
& \cos \theta =\cos \left( {{\tan }^{-1}}\frac{3}{4} \right)
\end{align}$
By the basic definition of the cosine function,
$\begin{align}
& \cos \theta =\frac{\text{side adjecent to angle }\theta }{\text{hypotenuse}} \\
& \cos \left( {{\tan }^{-1}}\frac{4}{3} \right)=\frac{4}{5}
\end{align}$
Hence, the value of $\cos \left( {{\tan }^{-1}}\frac{3}{4} \right)$ is $\frac{4}{5}$
