Solution
= Solution
The $n$th <Chebyshev polynomial> is defined on $[-1,1]$ by
$$
T_n(x)=\cos(n\arccos x).
$$
The cosine addition formula gives
$$
T_0(x)=1,
\qquad T_1(x)=x,
\qquad T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).
$$
Induction therefore shows that every $T_n$ is a <polynomial>. Its defining cosine formula also gives
$$
\boxed{-1\leq T_n(x)\leq1\quad(-1\leq x\leq1)}.
$$