There's a way to do it, which was the answers in the back of the book.
Let m = sin^-1 x
sin m = x
Now
cos^2 m = 1-sin^2 m
cos^2 m = 1 - x^2
Therefore:
cos m = sqrt (1-x^2)
m = cos^-1 sqrt (1-x^2)
therefore
sin^-1 x = cos^-1 sqrt (1-x^2)
What does QED stand for :S ?