Solve this

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.

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