Question

lf n is an integer then show that (1+i) 2n+(1-i) 2n =2n±¹cos npie/2

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Answer 1

Answer:

$1 + i = \sqrt{2} (cos \frac{\pi}{4} + i \: sin \frac{\pi}{4} )$

$1 - i = \sqrt{2} (cos \frac{\pi}{4} - i \: sin \frac{\pi}{4} )$

LHS,

$( \sqrt{2} (cos \frac{\pi}{4} + i \: sin \frac{\pi}{4} )) {}^{2n} + (\sqrt{2} (cos \frac{\pi}{4} - i \: sin \frac{\pi}{4} )) {}^{2n} =$

$( \sqrt{2} ) {}^{2n} (cos \frac{2n\pi}{4} + i \: sin \frac{2n\pi}{4} + cos \frac{2n\pi}{4} - i \: sin \frac{2n\pi}{4} ) =$

${2}^{n} \times 2 \: cos \frac{n\pi}{2} = {2}^{n + 1} \: cos \frac{n\pi}{2}$

Proved

Answer 2

Answer:

Excellent answer। nice work in above solution