Past exam of the mathematical methods I course of the University of Oxford / 2013 / 10 / c / Solution

Set $q=e^{i\theta }/2$. The required sine sum is the imaginary part of a finite geometric series:The denominator is $(5-4\cos \theta )/4$. The numerator's imaginary part isDividing and multiplying numerator and denominator by $2^N$ provesFor real $\theta$, $5-4\cos \theta \ge 1$, so there is no exceptional angle requiring a limiting interpretation. At $N=1$ the expression reduces to $\sin \theta /2$.