Find the simplified form of
${\cos ^{ – 1}}\left( {\frac{3}{5}\cos x + \frac{4}{5}\sin x} \right)$, where $x \in \left[ {\frac{{ – 3\pi }}{4},\frac{\pi }{4}} \right]$.
[NCERT,Exemplar.2.3,Q.13,Page.36]

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$

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