How do you simplify ##Sin(Cos^-1 x)##?

##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)##

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