The polynomials Pn(x) are what we need. Indeed, let α vary over [0,π]. Then
nα varies over [0,nπ], and the functions x =2cos α and Pn(x)=2cos nα range
over the segment [−2, 2]. Moreover, x covers this segment exactly once, while
Pn(x)coversit n times, assuming alternating values ±2for x = arccos(kπ/n),k =
0,...,n. This means that the graph of the polynomial Pn(x) lies in the strip |y|≤ 2
and contains n + 1 points of its alternating boundaries.
Let us summarize: there is a unique monic polynomial of degree n, given by