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MISCELLANEOUS EXERCISE 1 (II) · Q239

Q.If x1−y2+y1−x2=1x\sqrt{1-y^2}+y\sqrt{1-x^2}=1, then show that dydx=−1−y21−x2\dfrac{dy}{dx}=-\sqrt{\dfrac{1-y^2}{1-x^2}}

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Put x=sin⁡α, y=sin⁡βx=\sin\alpha,\ y=\sin\beta with α=sin⁡−1x, β=sin⁡−1y\alpha=\sin^{-1}x,\ \beta=\sin^{-1}y. Then 1−y2=cos⁡β\sqrt{1-y^2}=\cos\beta and 1−x2=cos⁡α\sqrt{1-x^2}=\cos\alpha, so the given equation becomes

x1−y2+y1−x2=sin⁡αcos⁡β+cos⁡αsin⁡β=sin⁡(α+β)=1.x\sqrt{1-y^2}+y\sqrt{1-x^2}=\sin\alpha\cos\beta+\cos\alpha\sin\beta=\sin(\alpha+\beta)=1.

Since sin⁡(α+β)=1\sin(\alpha+\beta)=1 and α,β∈[−π2,π2]\alpha,\beta\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right], we get α+β=π2\alpha+\beta=\dfrac{\pi}{2}, i.e.

sin⁡−1x+sin⁡−1y=π2.\sin^{-1}x+\sin^{-1}y=\frac{\pi}{2}. …

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