Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 2G / c / Solution

For $x=\cos \theta \in [-1,1]$,Both sides are polynomials in $x$. Since they agree on an interval, the polynomial identity theorem extends the equality to every real $x$. Thus $T_n$ is even for even $n$ and odd for odd $n$.