Question:
Solve $\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1} \frac{x-y}{x+y}$ is equal to
(A) $\frac{\pi}{2}$
(B). $\frac{\pi}{3}$
(C) $\frac{\pi}{4}$
(D) $\frac{-3 \pi}{4}$
Solution:
$\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1} \frac{x-y}{x+y}$
$=\tan ^{-1}\left[\frac{\frac{x}{y}-\frac{x-y}{x+y}}{1+\left(\frac{x}{y}\right)\left(\frac{x-y}{x+y}\right)}\right]$ $\left[\tan ^{-1} y-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}\right]$
$=\tan ^{-1}\left[\frac{\frac{x(x+y)-y(x-y)}{y(x+y)}}{\frac{y(x+y)+x(x-y)}{y(x+y)}}\right]$
$=\tan ^{-1}\left(\frac{x^{2}+x y-x y+y^{2}}{x y+y^{2}+x^{2}-x y}\right)$
$=\tan ^{-1}\left(\frac{x^{2}+y^{2}}{x^{2}+y^{2}}\right)=\tan ^{-1} 1=\frac{\pi}{4}$
Hence, the correct answer is C