Question

If $y=\sqrt{\sin x+\cos x+\sqrt{\sin x+\cos x+\sqrt{\sin x+\cos x+...\infty }}}$ find the value of $\frac{dy}{dx}$.

Answer 1

Lets call $\sin x+\cos x$ as $z$

$⟹y=\sqrt{z+\sqrt{z+\sqrt{z...\infty }}}$

$⟹y=\sqrt{z+y}$

Squaring on both sides,

$y^2=z+y$

$y^2-y=z$

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

Differentiating on both wrt to $x$

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

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