proove that sin^-1(x) = 1/(sqrt 1- x^2)

Question

I've got that answer without been prooved by whatever identities used. kindly asked for the steps to proove that derivative.

Answer 1

Question: prove that sin^-1(x) = 1/(sqrt 1- x^2)

You want to show that the derivative of sin^(-1)(x) is 1/sqrt(1 - x^2).

Let y = sin^-1(x)

Then, x = sin(y)

Now take the derivative of both sides (wrt y)

dx/dy = cos(y)

dx/dy = sqrt(1 - sin^2(y))

dx/dy = sqrt(1 - x^2)

Now reciprocate both sides.

dy/dx = 1/sqrt(1 - x^2)