本题目来源于试卷: Trigonometric Functions，类别为 IB数学
[问答题]
Let $z=\cos \theta+\mathrm{i} \sin \theta $, for $-\frac{\pi}{4}<\theta<\frac{\pi}{4} $.
1. 1. Find $z^{3}$ using the binomial theorem.
2. Use de Moivre's theorem to show that $\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta $ and $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$ .
2. Hence show that $\frac{\sin 3 \theta-\sin \theta}{\cos 3 \theta+\cos \theta}=\tan \theta $.
3. Given that $\sin \theta=\frac{1}{3}$ , find the exact value of $\tan 3 \theta $.
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