本题目来源于试卷: Complex Numbers，类别为 IB数学
[问答题]
1. Find the roots of z^{1,9 ko+h,uxq eca95srt 6}y6nrz;(yit. ky0xotk, 6 4ts mv kw;*x=1 which satisfy the condition $0<\arg (z)<\frac{\pi}{2}$ , expressing your answer in the form $r e^{\mathrm{i} \theta}$ , where r, $\theta \in \mathbb{R}^{+}$ .
2. Let S be the sum of the roots found in part (a).
1. Show that $\operatorname{Re}(S)=\operatorname{Im}(S)$ .
2. By writing $\frac{\pi}{8}$ as $\frac{1}{2} \cdot \frac{\pi}{4}$ , find the value of $\cos \left(\frac{\pi}{8}\right)$ in the form $\frac{\sqrt{a+\sqrt{b}}}{c} $, where a, b and c are integers to be determined.
3. Hence, or otherwise, show that $S=\frac{1}{2}(\sqrt{2+\sqrt{2}}+\sqrt{2}+\sqrt{2-\sqrt{2}})(1+\mathrm{i})$ .
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