Chapter 7.13, Problem 22MP

Use Poisson’s formula (Problem 21b) and Problem 20 to show that ${\displaystyle \underset{-\infty }{\overset{\infty }{\sum }}\frac{{\sin }^{2}n\theta }{n^2}}=\pi \theta ,0<\theta <\pi$.

(This sum is needed in the theory of scattering of light in a liquid.) Hint: Consider $f(x)$ and $g(\alpha )$ as in Problem 20. Note that $f(2k\pi )=0$ except for $k=0$ if $a<\pi$ Put $\alpha =n,a=\theta$.