Question

If $\frac{1}{2}y={\tan }^{-1}\left[\frac{5\cos x-12\sin x}{12\cos x+5\sin x}\right]$, then $\frac{dy}{dx}=$

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

Answer 1

The correct option is C $-2$

Given, $\frac{1}{2}y={\tan }^{-1}\left[\frac{5\cos x-12\sin x}{12\cos x+5\sin x}\right]$

Therefore, $\frac{y}{2}={\tan }^{-1}\left[\frac{\frac{5}{12}-\tan x}{1+\frac{5}{12}\tan x}\right]$

$={\tan }^{-1}\left(\frac{5}{12}\right)-{\tan }^{-1}\left(\tan x\right)$

$={\tan }^{-1}\left(\frac{5}{12}\right)-x$
$\Rightarrow \frac{dy}{dx}=-2$