Question:
If $\mathrm{y}=\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\ldots \text { to } \infty}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin \mathrm{x}}{1-2 \mathrm{y}}$.
Solution:
Here,
$y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\cdots \text { to } \infty}}}$
$y=\sqrt{\cos x+y}$
On squaring both sides,
$y^{2}=\cos x+y$
Differentiating both sides with respect to $x$,
$2 y \frac{d y}{d x}=-\sin x+\frac{d y}{d x}$
$\frac{d y}{d x}(2 y-1)=-\sin x$
$\frac{d y}{d x}=-\frac{\sin x}{2 y-1}$
$\frac{d y}{d x}=\frac{\sin x}{1-2 y}$
Hence proved.