. ${\sin ^2}x + {\cos ^2}y = 1$

We are given that, ${\sin ^2}x + {\cos ^2}y = 1$
Differentiating (i) on both sides w.r.t. x, we get

$2\sin x\cfrac{d}{{dx}}(\sin x) + 2\cos y\cfrac{d}{{dx}}(\cos y) = 0$

$\Rightarrow$ $2\sin x\cos x + 2\cos y( - \sin y)\cfrac{{dy}}{{dx}} = 0$

$\Rightarrow$ $\sin 2x - \sin 2y\cfrac{{dy}}{{dx}} = 0 \Rightarrow \cfrac{{dy}}{{dx}} = \cfrac{{\sin 2x}}{{\sin 2y}}$