Sum of Two Inverse Sines

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$$\sin^{-1} x + \sin^{-1} y = \sin^{-1} \left( x\sqrt{1-y^2} + y\sqrt{1-x^2} \right) \quad \mathrm{if} \quad x^2+y^2 \leq 1 \quad \mathrm{or} \quad xy \leq 0$$
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