Question 3 - A-Level Maths

By considering the binomial expansion of \(\left( z + z ^ { - 1 } \right) ^ { 4 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that \(\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )\).
Use the substitution \(x = \sin \theta\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).