Solution
= Solution
With $x=\cos\theta$, $T_k(x)=\cos k\theta$, proving orthogonality. In the convention $f\sim c_0+\sum_{k\ge1}c_kT_k$, $c_0=\pi^{-1}\int_0^\pi f(\cos\theta)d\theta$ and $c_k=2\pi^{-1}\int_0^\pi f(\cos\theta)\cos k\theta d\theta$. For $f=\sqrt{1-x^2}$, $c_0=2/\pi$, odd coefficients vanish, and $c_{2m}=-4/[\pi(4m^2-1)]$.