Question

If $y=\tan ^{-1}\left[\frac{5 \cos x-12 \sin x}{12 \cos x+5 \sin x}\right]$, then $\frac{d y}{d x}$ is equal to

• A. 1
• B. -1
• C. -2
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

Given,
$y=\tan ^{-1}\left[\frac{5 \cos x-12 \sin x}{12 \cos x+5 \sin x}\right]$
$=\tan ^{-1}\left[\frac{\frac{5}{12}-\tan x}{1+\frac{5}{12} \tan x}\right]$
[divide by $12 \cos x$ in denominator and numerator]
$=\tan ^{-1}\left(\frac{5}{12}\right)-\tan ^{-1}(\tan x) $
$y=\tan ^{-1}\left(\frac{5}{12}\right)-x$
On differentiating both sides w.r.t. $x $, we get
$\frac{d y}{d x}=\frac{d}{d x}\left\{\tan ^{-1}\left(\frac{5}{12}\right)\right\}-\frac{d}{d x}(x)=0-1$
$\therefore \frac{d y}{d x}=-1$