Question 1
Let $\alpha$ vary freely and put $x=\cos\alpha$, $y=\cos n\alpha$.
(a) Prove that each $x$ in $-1\le x\le1$ determines exactly one value of $y$. Regard $y$ as a function $T_n(x)$. Calculate $T_1(x),T_2(x)$ and prove
$$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).$$
Deduce that $T_n$ is a polynomial of degree $n$.
(b) Prove that $T_n$ has $n$ distinct real roots in $[-1,1]$.
(a) Prove that each $x$ in $-1\le x\le1$ determines exactly one value of $y$. Regard $y$ as a function $T_n(x)$. Calculate $T_1(x),T_2(x)$ and prove
$$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).$$
Deduce that $T_n$ is a polynomial of degree $n$.
(b) Prove that $T_n$ has $n$ distinct real roots in $[-1,1]$.