本题目来源于试卷: Complex Numbers，类别为 IB数学
[问答题]
1. 1. Expand $(\cos \theta+\mathrm{i} \sin \theta)^{4}$ by using the binomial theorem.
2. Hence use de Moivre's theorem to prove that
$\cos 4 \theta=\cos ^{4} \theta-6 \cos ^{2} \theta \sin ^{2} \theta+\sin ^{4} \theta$
3. State a similar expression for $\sin 4 \theta $ in terms of $\cos \theta $and $ \sin \theta$ .
Let $ z=r(\cos \alpha+\mathrm{i} \sin \alpha)$ , where $\alpha$ is measured in degrees, be the solution of $z^{4}-\mathrm{i}=0$ which has the smallest positive argument.
2. Find the modulus and argument of z .
3. Use (a) (ii) and your answer from (b) to show that $8 \cos ^{4} \alpha-8 \cos ^{2} \alpha+1=0$ .
4. Hence express $\cos 22.5^{\circ}$ in the form $\frac{\sqrt{a+b \sqrt{c}}}{d}$ where a, b, c, d $\in \mathbb{Z}$ .
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