VMO 1972 Problem 1

Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ( $n \in Z$ ). i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$ . Thus we can write $y$ as a function of $x, y = T_n(x)$ . Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ . From this it follows that $T_n(x)$ is a polynomial of degree $n$ . ii) Prove that the polynomial $T_n(x$ ) has $n$ distinct roots in $[-1, 1]$ .