= Solution
Using the normalized <eigenfunctions> and the same change of variables, their <weighted inner product> is
$$
\int_{-1}^1\frac{y_m(x)y_n(x)}{\sqrt{1-x^2}}\,dx=\frac2\pi\int_0^\pi\sin(m\theta)\sin(n\theta)\,d\theta.
$$
For distinct positive integers $m,n$, the product-to-sum identity gives
$$
\frac1\pi\int_0^\pi[\cos((m-n)\theta)-\cos((m+n)\theta)]d\theta=0,
$$
because each integrated sine has integer-multiple-of-$\pi$ endpoints. For $m=n$, the integral is one by the normalization already calculated. Hence
$$
\boxed{\int_{-1}^1w(x)y_m(x)y_n(x)\,dx=\delta_{mn}.}
$$
This verifies <orthonormality> explicitly, with $\delta_{mn}$ the <Kronecker delta>.
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