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Multiplication by $(1-x^2{)}^{-1/2}$ puts the equation in Sturm-Liouville theory form:Multiply the equations for $T_n,T_m$ by $T_m,T_n$, subtract and integrate. The boundary term vanishes because $\sqrt{1-x^2}=0$ at both endpoints and the polynomial derivatives are bounded. Therefore, for $n^2\ne m^2$,This is the usual Chebyshev polynomial orthogonality.

Differentiating the original equation and writing $U_n=T_n^{\prime }$ givesIts self-adjoint form isThe same subtraction argument now yields, for $n^2\ne m^2$,These two relations form the Chebyshev derivative Sturm-Liouville pair.