If $\sin^{-1}x + \sin^{-1}y=\sin^{-1}(x \sqrt{1-y^{2}}+ y \sqrt{1-x^{2}})$, then what is the area represented by the locus of point $(x, y)$? I'm totally blank about this question so please explain clearly and state all the basic steps!
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See http://math.stackexchange.com/questions/672575/proof-for-the-formula-of-sum-of-arcsine-functions-arcsin-x-arcsin-y – lab bhattacharjee Jun 01 '16 at 07:18
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$$\alpha=sin^{-1}x\to sin\alpha=x \,\,\,\,\,and\,\,cos\,\alpha=\sqrt{1-sin^2 \alpha}=\sqrt{1-x^2}$$ $$\beta=sin^{-1}y\to sin\beta=y \,\,\,\,\,and\,\,cos\,\beta=\sqrt{1-sin^2\beta}=\sqrt{1-y^2}$$ let $t=sin^{-1}x+sin^{-1}y=\alpha+\beta$ , then
$$sin(t)=sin(\alpha+\beta)=sin(\alpha)cos(\beta)+cos(\alpha)sin(\beta)$$ $$sin(t)=x\sqrt{1-y^2}+\sqrt{1-x^2}y$$ $$t=sin^{-1}(x\sqrt{1-y^2}+\sqrt{1-x^2}y)$$
Behrouz Maleki
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Now the question is finally fully edited. Please provide the detailed explanation to the method of solving this problem – Ujjwal May 31 '16 at 17:06
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That got me back to the same old problem. Please explain how to proceed and what about the area? – Ujjwal Jun 01 '16 at 01:26
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