Solution

We have,

\[ \cos^{-1}\left(\frac{3}{5}\cos x + \frac{4}{5}\sin x\right) \]

Let \[ \cos y = \frac{3}{5} \quad \Rightarrow \quad \sin y = \frac{4}{5} \]

Therefore, \[ y = \cos^{-1}\frac{3}{5} = \sin^{-1}\frac{4}{5} = \tan^{-1}\left(\frac{4}{3}\right) \]

Now, \[ \cos^{-1}(\cos y \cos x + \sin y \sin x) \]

\[ = \cos^{-1}(\cos(y – x)) \]

\[ = y – x \]

\[ \boxed{\tan^{-1}\left(\frac{4}{3}\right) – x} \]