I was reading on Chebyshev functions, and I found lots of resources on proving the orthogonality of Chebyshev polynomials of the first kind:
$\int_{-1}^1 T_m(x) T_n(x) \frac{dx}{\sqrt{1-x^2}} = \begin{cases} 0 & \text{if $m\neq n$} \\ \pi & \text{if $m=n=0$} \\ \pi/2 & \text{if $m=n\neq 0$} \end{cases}$
But I've found no resources on proving the orthogonality for polynomials of the second kind. The resources I've read only say "similarly, we find that:"
$\int_{-1}^1 U_m(x) U_n(x) \sqrt{1-x^2} dx = \begin{cases} 0 & \text{if $m\neq n$} \\ \pi/2 & \text{if $m=n$} \end{cases}$
Does anyone know how to show the above orthogonality explicitly?