Question

Write the given expression in terms of x and y only. sin(sin^−1 (x) + cos^−1 (y))

Answer 1

$sin(A+B)=sinAcosB+sinBcosA \\ sin(sin^{-1} x+cos^{-1} y) \\ sin(cos^{-1} y)= \sqrt{1- y^{2} } \\ cos(sin^{-1} x)= \sqrt{1- x^{2} } \\ =sin(sin^{-1} x)cos(cos^{-1} y)+sin(cos^{-1} y)cos(sin^{-1} x) \\ =xy+\sqrt{1- y^{2} } \sqrt{1- x^{2}} \\ =xy+ \sqrt{(1- y^{2})(1- x^{2})}$