Question

If $y={\cos }^{-1}\left(\frac{5\cos x-12\sin x}{13}\right),xϵ(0,\frac{\pi }{2})$, then $\frac{dy}{dx}$ is equal to

• A. $1$
• B. $-1$
• C. $0$
• D. none of these

Answer 1

The correct option is A $1$

$y={\cos }^{-1}\left(\frac{5\cos x-12\sin x}{13}\right)$

$y={\cos }^{-1}(\frac{5}{13}\cos x-\frac{12}{13}\sin x)$

Let $\cos \varphi =\frac{5}{13}$ and $\sin \varphi =\frac{12}{13}$

So, $y={\cos }^{-1}\left(cos\right(x+\varphi \left)\right)$

$\Rightarrow y=x+\varphi$

$\Rightarrow \frac{dy}{dx}=1$