Past exam of the mathematical methods I course of the University of Oxford / 2019 / 9 / a / Solution

Put $w=e^{i\theta }$. By Euler's formula, $w-w^{-1}=2i\sin \theta$. The binomial theorem givesGroup opposite powers and use $w^j-w^{-j}=2i\sin (j\theta )$:Since $(2i{)}^7=-128i$, division by $-2i$ yieldsThis derivation is an application of the same exponential representation underlying De Moivre's theorem.