Let A = sin-1x.
Then, we have
sinA = x
Already we know the trigonometric identity
sin2A + cos2A = 1
x2 + cos2A = 1
cos2A = 1 - x2
Let B = sin-1y.
Then, we have
sinB = y
sin2B + cos2B = 1
y2 + cos2B = 1
cos2B = 1 - y2
We know the following compound angle formula.
sin(A + B) = sinAcosB + cosAsinB
Substitute.
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