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))}`