Question

If $y=\frac{\sin x}{1+\frac{\cos x}{1+\frac{\sin x}{1+\frac{\cos x}{1+.....\infty }}}},$
then $\frac{dy}{dx}$ at $x=\frac{\pi }{2}$ is

Answer 1

$y=\frac{\sin x}{1+\frac{\cos x}{1+y}}$

$\Rightarrow y=\frac{(1+y)\sin x}{1+y+\cos x}$

$\Rightarrow y+y^2+y\cos x=\sin x+y\sin x$

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

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

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

At $x=\frac{\pi }{2}$

$\therefore \frac{dy}{dx}=\frac{y}{2y}=\frac{1}{2}=0.5$