Question

If ${\sin }^{2}x+{\cos }^{2}y=1$ find $\frac{dy}{dx}$

Answer 1

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

$\Rightarrow {\sin }^{2}x=1-{\cos }^{2}y$

$\Rightarrow {\sin }^{2}x={\sin }^{2}y$

$\Rightarrow {\sin }^{2}y={\sin }^{2}x$

Differentiating both sides, w.r.t $x$ we get

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

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