Chapter 2.16, Problem 12P

In optics, the following expression needs to be evaluated in calculating the intensity of light transmitted through a film after multiple reflections at the surfaces of the film:

${\left({\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{r}^{2n}\cos n\theta }\right)}^2+{\left({\displaystyle \underset{n=o}{\overset{\infty }{\sum }}{r}^{2n}\sin n\theta }\right)}^2$

Show that this is equal to and so evaluate it assuming |r| < 1 (r is the fraction of light reflected each time).