If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + ... +∞}}}$ then $\frac{dy}{dx}$ equals to

$\frac{\cos x}{2y-1}$

The correct answer is Option (2) → $\frac{\cos x}{2y-1}$

Let $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots}}}$

Then, $y = \sqrt{\sin x + y}$

Squaring both sides: $y^2 = \sin x + y$

$\Rightarrow y^2 - y - \sin x = 0$

Differentiate both sides: $2y \cdot \frac{dy}{dx} - \frac{dy}{dx} - \cos x = 0$

$\Rightarrow \frac{dy}{dx} (2y - 1) = \cos x$

$\Rightarrow \frac{dy}{dx} = \frac{\cos x}{2y - 1}$