Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\left ( \frac{1+i}{1-i}\right )^{30} +\left ( \frac{1-i}{1+i}\right )^{30} =$	A. 1
	B. 2
	C. 0
	D. − 2
	E.

$\left ( \frac{1+i}{1-i}\right )^{30} +\left ( \frac{1-i}{1+i}\right )^{30} \\ =\left ( \frac{(1+i)^2}{(1-i)(1+i)}\right )^{30} +\left ( \frac{(1-i)^2}{(1+i)(1-i))}\right )^{30}\\ =\left ( \frac{1+2i+i^2}{1-i^2}\right )^{30} +\left ( \frac{1-2i+i^2}{1-i^2}\right )^{30}=\left ( \frac{2i}{2} \right )^{30}+\left ( \frac{-2i}{2} \right )^{30}\\ =i^{30}+i^{30}=2i^2=-2$