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If n is a positive integer than show that ( 1+i ) ^{ 2n }+( 1-

Question

If n is a positive integer than show that
$(1 + i)^{2 n} + (1 - i)^{2 n} = 2^{(n + 1)} cos (nπ /2)$

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If n is a positive integer than show that

(1 + i)^{2 n} + (1 - i)^{2 n} = 2^{(n + 1)} cos (nπ /2)

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Text solutionVerified

(1 + i) = 2 (cos \frac{π}{4} + s in \frac{π}{4})

1 - i = 2 (cos \frac{π}{4} - s in \frac{π}{4})

(1 + i)^{2 n} = (2)^{2 n} (cos (\frac{2 nπ}{4}) + i s in (\frac{2 nπ}{4}))

(1 - i)^{2 n} = (2)^{2 n} [cos (\frac{2 nπ}{4}) - i s in (\frac{2 nπ}{4})]

Adding

(1 + i)^{2 n} + (1 - i)^{2 n} = (2)^{2 n} [2 cos (\frac{2 nπ}{4})]

= 2^{\frac{2 n}{n}} . 2^{1} cos (\frac{nπ}{2})

= 2^{n + 1} cos (\frac{2 nπ}{2})

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Question Text	If n is a positive integer than show that $(1 + i)^{2 n} + (1 - i)^{2 n} = 2^{(n + 1)} cos (nπ /2)$
Answer Type	Text solution:1
Upvotes	150