Find dy dx if ^2 x + ^2 y = 1 - Vidyadip

Question

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

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Answer

It is given that $\sin^2 x + \cos^2 y = 1$
Differentiating both sides $\text{w.r.t x}$, we get,
$\frac{d}{d x}\left(\sin ^{2} x+\cos ^{2} y\right)=\frac{d}{d x}(1)$
$\Rightarrow \frac{d}{d x}\left(\sin ^{2} x\right)+\frac{d}{d x}\left(\cos ^{2} y\right)=0$
$\Rightarrow 2 \sin x \cdot \frac{d}{d x}(\sin x)+2 \cos y \cdot \frac{d}{d x}(\cos y)=0$
$\Rightarrow 2 \sin x \cos x+2 \cos y(-\sin y) \cdot \frac{d y}{d x}=0$
$\Rightarrow \sin 2 x-\sin 2 y \frac{d y}{d x}=0$
$\Rightarrow \frac{d y}{d x}=\frac{\sin 2 x}{\sin 2 y}$

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