Given the definition of Chebyshev polynomials in this form:
$$T_n(x) = \cos(n\cos^{-1}x), n\ge 1, T_0=1$$
I want to show that using Gram-Schmidt procedure with set $\{1, x, x^2, \dots\}$ and weight function $\omega = (1-x^2)^{-1/2}$ generates this sequence $\{T_n\}$.
Also, I used de Moivre's Identity and found an equivalent form:
$$T_n(x) = x^n + \binom{n}{2}x^{n-2}(x^2-1) + \binom{n}{4}x^{n-4}(x^2-1)^2 + \dots$$
However I cannot go from Gram-Schmidt procedure to these forms. Gram-Schmidt yields a linear combination of computed Chebyshev polynomials and the next generating element, but I can't arrange the coefficients since they seem quite random to me.