Prove De Moivre’s theorem for powers of positive integers.

Let \(z = r(\cos(\theta) + i\sin(\theta))\) where \(r > 0\) and \(\theta \in \mathbb{R}\). (a) Verify that De Moivre's theorem holds for \(n = 2\) by expanding \(z^2\) and showing that \(z^2 = r^2(\cos(2\theta) + i\sin(2\theta))\). (1 mark) (b) State the trigonometric identity used to simplify your result in part (a). (1 mark)

Use mathematical induction to prove De Moivre's theorem: for all positive integers \(n\), \((\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)\).

When using proof by mathematical induction to prove De Moivre's theorem \((r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))\) for \(n \in \mathbb{Z}^+\), the inductive step requires proving

Use mathematical induction to prove that \((\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)\) for all \(n \in \mathbb{Z}^+\).