Question 11 - A-Level Maths

De Moivre's theorem states that $\quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$ for $n \in \Re$
1. Use induction to prove de Moivre's theorem for $n \in \mathbb { Z } ^ { + }$.
2. Show that $\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$
3. Hence show that $2 \cos \frac { \pi } { 10 }$ is a root of the equation
$$x ^ { 4 } - 5 x ^ { 2 } + 5 = 0$$