sin(cos^(-1)(x)) = sqrt(1-x^2) Let's draw a right triangle with an angle of a = cos^(-1)(x). As we know cos(a) = x = x/1 we can label the adjacent leg as x and the hypotenuse as 1. The Pythagorean theorem then allows us to solve for the second leg as sqrt(1-x^2). With this, we can now find sin(cos^(-1)(x)) as the quotient of the opposite leg and the hypotenuse. sin(cos^(-1)(x)) = sin(a) = sqrt(1-x^2)/1 = sqrt(1-x^2)
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