Chebyshev polynomials $T_{n}(x)$ satisfy the differential equation

$$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+n^{2} y=0 \quad \text { on } \quad[-1,1],$$

where $n$ is an integer.

Recast this equation into Sturm-Liouville form and hence write down the orthogonality relationship between $T_{n}(x)$ and $T_{m}(x)$ for $n \neq m$.

By writing $x=\cos \theta$, or otherwise, show that the polynomial solutions of ( $\dagger$ ) are proportional to $\cos \left(n \cos ^{-1} x\right)$.