Consider \(f(x)=4 \sin ^{3} 2 x\).

a. Derive the trigonometric identity giving \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta\) using DeMoivre's Formula.

b. Find the Fourier series of \(f(x)=4 \sin ^{3} 2 x\) on \([0,2 \pi]\) without computing any integrals.