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  • If x sin (a + y) + sin a cos (a + y) = 0, prove that dy/dx = root 1-y^2/1-x^2

If x \sin (a + y) + \sin a \cos (a + y) = 0, prove that \frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}.

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Given: x \sin (a + y) + \sin a \cos (a + y) = 0

\begin{aligned} &\Rightarrow x=\frac{-\sin a \cos (a+y)}{\sin (a+y)}\\ &\Rightarrow x=-\sin a \cot (a+y)\left[\because \frac{\cos x}{\sin x}=\cot x\right]\\ &\text { Differentiating with respect to } y \text { , we get }\\ &\frac{d x}{d y}=\frac{d}{d y}[-\sin a \cot (a+y)]\\ &\Rightarrow \frac{d x}{d y}=-\sin a \frac{d}{d y} \cot (a+y) \end{aligned}

\\ \Rightarrow \frac{d x}{d y}=-\sin a\left[-\operatorname{cosec}^{2}(a+y)\right] \because\left[\frac{d}{d y} \cot (x)=\operatorname{cosec}^{2} x\right] \\ \Rightarrow \frac{d x}{d y}=\sin a\left[\frac{1}{\sin ^{2}(a+y)}\right] \\ \Rightarrow \frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}

Hence Proved.

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