2022 YIJC P1 Q6

Show that \(1+{{\mathrm{e}}^{-\mathrm{i}\alpha }}=2\cos \frac{\alpha }{2}{{\mathrm{e}}^{-\mathrm{i}\frac{\alpha }{2}}}\), where \(-\pi <\alpha \le \pi \).[2]

Hence or otherwise, show that \({{\left( 1+{{\mathrm{e}}^{-\mathrm{i}\alpha }} \right)}^{3}}-{{\left( 1+{{\mathrm{e}}^{\mathrm{i}\alpha }} \right)}^{3}}=-16\mathrm{i}{{\cos }^{3}}\left( \frac{\alpha }{2} \right)\sin \left( \frac{3\alpha }{2} \right)\).[3]

Given further that \(0<\alpha <\frac{2}{3}\pi \) and \(z={{\left( 1+{{\mathrm{e}}^{-\mathrm{i}\alpha }} \right)}^{3}}-{{\left( 1+{{\mathrm{e}}^{\mathrm{i}\alpha }} \right)}^{3}}\), deduce the modulus and argument of \(z\). Express your answers in terms of \(\alpha \) whenever applicable.[2]