Exercise 15, § 1

Let $u$ and $v$ be two polynomials in $x$, with real coefficients, such that $\sqrt{1-u^2} = v \sqrt{1-x^2}$ identically; show that, if $n$ is the degree of $u$, then $u' = nv$; deduce that $u(x) = \cos(n \operatorname{Arc cos} x)$.