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Real Analysis Problem

The inverse cosine function has domain [-1,1]and range [0, pi]. Prove that (cos^-1)'(x) = -1/ sqrt(1-x^2).

This needs to be proved from a real analysis point of view not a calculus.

Solution Preview

Let y = cos^-1(x)
<br>So now we need to prove that dy/dx = -1/sqrt(1-x^2)
<br>Let x = cos(t)------eqn 1
<br>Therefor y = cos^-1(cos(t))
<br> or y = t (because cos^-1 of cos means just ...