Question

If $y={\tan }^{-1}\left[\frac{5\cos x-12\sin x}{12\cos x+5\sin x}\right]$, then $\frac{dy}{dx}$ 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
$\frac{dy}{dx}=\frac{d}{dx}\left\{{\tan }^{-1}\left(\frac{5}{12}\right)\right\}-\frac{d}{dx}(x)=0-1$
$\therefore \frac{dy}{dx}=-1$