<s>[INST] 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 $(x, y)$? 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!
 [/INST] $$\alpha=sin^{-1}x\to sin\alpha=x \,\,\,\,\,and\,\,cos\,\alpha=\sqrt{1-sin^2 \alpha}=\sqrt{1-x^2}$$
 $$\beta=sin^{-1}y\to sin\beta=y \,\,\,\,\,and\,\,cos\,\beta=\sqrt{1-sin^2\beta}=\sqrt{1-y^2}$$
let $t=sin^{-1}x+sin^{-1}y=\alpha+\beta$ , then 
$$sin(t)=sin(\alpha+\beta)=sin(\alpha)cos(\beta)+cos(\alpha)sin(\beta)$$
$$sin(t)=x\sqrt{1-y^2}+\sqrt{1-x^2}y$$
$$t=sin^{-1}(x\sqrt{1-y^2}+\sqrt{1-x^2}y)$$
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