In the proof of Lemma 2 of Driscoll and Healy, it says \begin{align} \sqrt{2}\delta_{k,0} &= \frac{1}{\sqrt{2}}\int_0^\pi P_k(\cos\theta)\sin\theta d\theta\\\\ &= \frac{1}{2\sqrt{2}}\int_{-\pi}^\pi P_k(\cos\theta)\sin\theta \prod_n sgn(\theta) d\theta\\\\ &= \frac{1}{2\sqrt{2}}\frac{\pi}{n}\sum_{l=-n}^{n-1} P_k(\cos\frac{l\pi}{n})\sin(\frac{l\pi}{n}) \prod_n sgn(\frac{l\pi}{n})\\\\ &= \frac{\pi}{n\sqrt{2}}\sum_{l=0}^{n-1} P_k(\cos\frac{l\pi}{n})\sin(\frac{l\pi}{n}) \prod_n sgn(\frac{l\pi}{n}). \end{align} I don't understand how to go from the second equality to the third equality. I believe it uses the discrete orthogonality of sines and cosines and the fact that $\sum_{l=-n}^{n-1}\cos(\frac{\pi kl}{n}) = \frac{n}{\pi}\int_{-\pi}^\pi \cos(k\theta)$ and $\sum_{l=-n}^{n-1}\sin(\frac{\pi kl}{n}) = \frac{n}{\pi}\int_{-\pi}^\pi \sin(k\theta),$ but it is not clear to me why the third equality is true since it is an integral of the product of functions of sines and cosines. Can someone explain this to me or point me the right resources?
