SOLUTION : Let $y=\tan ^{-1}\left[\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right]=\tan ^{-1}\left[\frac{\frac{a \cos x}{b \cos x}-\frac{b \sin x}{b \cos x}}{b \cos x}+\frac{a \sin x}{a \cos x}\right]=\tan ^{-1}\left[\frac{\frac{a}{b}-\tan x}{1+\frac{a}{b} \tan x}\right]$
$=\tan ^{-1} \frac{a}{b}-\tan ^{-1}(\tan x)\left[\because \tan ^{-1}\left(\frac{x-y}{1+x y}\right)=\tan ^{-1} x-\tan ^{-1} y\right]=\tan ^{-1} \frac{a}{b}-x$
