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Question

Show that: Is n ∈ ℤ, than
$((1+i)^n + (1-i)^n)/2 = (sqrt(2))^n * cos (n * 45°)$
Thank you!

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Answer 1 · 2018-01-14T14:08:48

See below.

We know that

$x+i y = sqrt(x^2+y^2) e^(i phi)$ with $phi = arctan(y/x)$ then

$(1+i) = sqrt2 e^(i phi)$
$(1-i) = sqrt2 e^(-iphi)$ then

$((1+i)^n+(1-i)^n)/2 = (sqrt2)^n((e^(i nphi)+e^(-i n phi))/2) =$

$= (sqrt2)^n cos(n phi)$ but $phi = arctan 1 = 45^@$ then

$((1+i)^n+(1-i)^n)/2 = (sqrt2)^n cos(n 45^@)$