Find the simplified form of
${\cos ^{ - 1}}\left( {\frac{3}{5}\cos x + \frac{4}{5}\sin x} \right)$, where $x \in \left[ {\frac{{ - 3\pi }}{4},\frac{\pi }{4}} \right]$.

We have, ${\cos ^{ - 1}}\left[ {\frac{3}{5}\cos x + \frac{4}{5}\sin x} \right],x \in \left[ {\frac{{ - 3\pi }}{4},\frac{\pi }{4}} \right]$

Let $\cos y = \frac{3}{5}$
$\Rightarrow$ $\sin y = \frac{4}{5}$

$\Rightarrow$ $y = {\cos ^{ - 1}}\frac{3}{5} = {\sin ^{ - 1}}\frac{4}{5} = {\tan ^{ - 1}}\left( {\frac{4}{3}} \right)$

$therefore,$ ${\cos ^{ - 1}}[\cos y \cdot \cos x + \sin y \cdot \sin x]$
$= {\cos ^{ - 1}}[\cos (y - x)]$

$= y - x = {\tan ^{ - 1}}\frac{4}{3} - x$