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Algebra of Complex Numbers

Compute sin...

Question

Compute ${⎛ ⎜ ⎜ ⎝ \frac{sin \frac{π}{8} + i cos \frac{π}{8}}{sin \frac{π}{8} - i cos \frac{π}{8}} ⎞ ⎟ ⎟ ⎠}^{8}$

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Solution

Let $z = {⎛ ⎜ ⎜ ⎝ \frac{sin \frac{π}{8} + i cos \frac{π}{8}}{sin \frac{π}{8} - i cos \frac{π}{8}} ⎞ ⎟ ⎟ ⎠}^{8}$
i.e. $z = \frac{{(sin \frac{π}{8} + i cos \frac{π}{8})}^{8}}{{(sin \frac{π}{8} - i cos \frac{π}{8})}^{8}}$
$z = \frac{i^{8} {(cos \frac{π}{8} - i sin \frac{π}{8})}^{8}}{{(- i)}^{8} {(cos \frac{π}{8} + i sin \frac{π}{8})}^{8}}$
$z = \frac{cos π - i sin π}{cos π + i sin π} = 1$
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