Question

If $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\dots \infty }}}$, then $(2y-1)\frac{dy}{dx}$ is equal to

• A. $\sin x$
• B. $-\cos x$
• C. $\cos x$
• D. $-\sin x$

Answer 1

The correct option is C $\cos x$

Given, $y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\dots \infty }}}$
$\therefore y=\sqrt{\sin x+y}$
$\Rightarrow y^2=\sin x+y$
On differentiating with respect to $x$, we get
$2y\frac{dy}{dx}=\cos x+\frac{dy}{dx}$
$\Rightarrow (2y-1)\frac{dy}{dx}=\cos x$