9758/2021/P1/Q04

Do not use a calculator in answering this question.
The complex number $z$ is given by

$z=\frac{\left(\cos \left(\frac{1}{16} \pi\right)+i \sin \left(\frac{1}{16} \pi\right)\right)^2}{\cos \left(\frac{1}{8} \pi\right)-i \sin \left(\frac{1}{8} \pi\right)}$.

(a) Find $|z|$ and $\arg (z)$. Hence find the value of $z^2$.

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(b) (i) Show that
$(\cos \theta+i \sin \theta)(1+\cos \theta-i \sin \theta)=1+\cos \theta+i \sin \theta$

[2]

(ii) Hence, or otherwise, find the value of $(1+z)^4+\left(1+z^*\right)^4$.

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