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I am familiar with the following integral for a quotient of squares of sines in terms of the Bessel function of the first kind $J$: $$ \int_{0}^{\pi}\frac{\sin^{2}\left(\frac{N}{2}\cos(\theta)\right)}{\sin^{2}\left(\frac{1}{2}\cos(\theta)\right)}\,d\theta=\pi\left(N+2\sum_{n=1}^{N-1}(N-n)J_{0}(n)...