Past exam of the mathematical methods I course of the University of Oxford / 2022 / 9 / b

Use the imaginary part of the finite complex sum:The identity $1+e^{i\theta }=2e^{i\theta /2}\cos (\theta /2)$ then givesHere $N$ is a nonnegative integer. Keeping the signed cosine raised to the integer power makes the result valid even when $\cos (\theta /2)<0$; inserting an absolute value would lose the sign for odd $N$.