|COMPLEXE NUMBERS| Show that...? Thank you!

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^@)#