If n is a positive integer, then (1 + i)^n + (1 - i)^n is equal t… - Vidyadip

MCQ

If $n$ is a positive integer, then ${(1 + i)^n} + {(1 - i)^n}$ is equal to

A
${(\sqrt 2 )^{n - 2}}\cos \left( {\frac{{n\pi }}{4}} \right)$
B
${(\sqrt 2 )^{n - 2}}\sin \left( {\frac{{n\pi }}{4}} \right)$
✓
${(\sqrt 2 )^{n + 2}}\cos \left( {\frac{{n\pi }}{4}} \right)$
D
${(\sqrt 2 )^{n + 2}}\sin \left( {\frac{{n\pi }}{4}} \right)$

✓

Answer

Correct option: C.

${(\sqrt 2 )^{n + 2}}\cos \left( {\frac{{n\pi }}{4}} \right)$

c
(c)${(1 + i)^n} + {(1 - i)^n}$
$ = {(2)^{n/2}}\left\{ {\cos \frac{{n\pi }}{4} + i\sin \frac{{n\pi }}{4} + \cos \frac{{n\pi }}{4} - i\sin \frac{{n\pi }}{4}} \right\}$
$ = \,{2^{\frac{n}{2}}}.\,2\cos \frac{{n\pi }}{4} = {2^{\frac{n}{2} + 1}}\cos \frac{{n\pi }}{4} = {(\sqrt 2 )^{n + 2}}\cos \frac{{n\pi }}{4}$.

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