I computed the term $(T_{3})$ in the Chebyshev polynomials on Wolfram Alpha:
After viewing this integral, I am wondering if each polynomial in the Chebyshev polynomial sequence, $T_{n}$, is orthogonal with respect to the weight function $(1-x^{2})^{-1/2}$. If so, is that the reason why each product of two different Tchebychev polynomials are orthogonal to eachother with respect to this weight function?
Indeed, Chebyshev polynomials are orthogonal with respect to the $\sqrt{1-x^2}^{-1}.$ The "reason" behind it is that the sequence $\cos{nx},$ $n\ge 1$ is orthogonal on $[0,2\pi].$ More precisely, recall that $T_n(\cos x)=\cos{nx}$ and thus for $m\ne n,$ making change of variables in the integral below leads to $$\int_{-1}^1T_n(x)T_m(x)\frac{1}{\sqrt{1-x^2}}dx=\int_{0}^{\pi}\cos{ny}\cdot\cos{my}dy=0.$$
For more information see wikipedia page devoted to the Chebyshev polynomials: see here