##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##. 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)##