Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 11H / g / Solution

With $x=\cos \theta$, the weight becomes ${\sin }^2\theta d\theta$. Sincethe sine orthogonality relations show that $P_n$ is a scalar multiple of the Chebyshev polynomial of the second kind $U_n$. Its roots areThe printed normalization $P_n(0)=1$ is possible only for even $n$; then $U_n(0)=(-1{)}^{n/2}$ andFor odd $n$, every such orthogonal polynomial is odd and vanishes at zero, so that normalization is inconsistent.