Chapter 10.3, Problem 32E

The Chebyshev (Tchebichef) polynomials $T_n\left(x\right)$ are orthogonal on the interval $\left[-1,1\right]$ with respect to the weight function $w\left(x\right)={\left(1-x^2\right)}^{-\frac{1}{2}}$. Verify this fact for the first three Chebyshev polynomials:

$T_0\left(x\right)\equiv 1$, $T_1\left(x\right)=x$, $T_2\left(x\right)=2x^2-1$.