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

Take the imaginary part of the eighth-power binomial expansion in De Moivre's theorem. With $s=\sin \theta$ and $c=\cos \theta$,Substitute $c^2=1-s^2$ and collect powers:Therefore the requested leading coefficient isThe other coefficients, if retained as an algebraic check, are $b=192,c=-80,d=8$. No division by $s$ or $c$ is needed, so the identity remains valid when either vanishes.