Question

If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+...\mathrm{to}\infty }}}$, prove that $\frac{dy}{dx}=\frac{\sin x}{1-2y}$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+...to\infty }}} \\ \Rightarrow y=\sqrt{\cos x+y} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}, \\ {y}^2=\cos x+y \\ \Rightarrow 2y\frac{dy}{dx}=-\sin x+\frac{dy}{dx} \\ \Rightarrow \frac{dy}{dx}\left(2y-1\right)=-\sin x \\ \Rightarrow \frac{dy}{dx}=\frac{-\sin x}{2y-1} \\ \Rightarrow \frac{dy}{dx}=\frac{\sin x}{1-2y} \end{gathered}$$