Question 8 - A-Level Maths

Use De Moivre's Theorem to show that, if $z = \cos \theta + \mathrm { i } \sin \theta$, then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
1. Expand $\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 4 }$.
2. Show that $$\cos ^ { 4 } 2 \theta = A \cos 8 \theta + B \cos 4 \theta + C$$ where $A , B$ and $C$ are rational numbers.
Hence solve the equation $$8 \cos ^ { 4 } 2 \theta = \cos 8 \theta + 5$$ for $0 \leqslant \theta \leqslant \pi$, giving each solution in the form $k \pi$.
Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 4 } 2 \theta d \theta = \frac { 3 \pi } { 16 }$$