Show that #(1/sqrt2 + i/sqrt2)^10 + (1/sqrt2 - i/sqrt2)^10=0#?

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So how do I proceed? Just expand?

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Answer 1 · 2017-11-05T16:43:21

See below.

Using the de Miovre's identity

#e^(iphi) = cos phi+i sin phi# with #phi = pi/4# we have

#1/sqrt(2)(1+i) = e^(i pi/4)# and it's conjugate

#1/sqrt(2)(1-i) = e^(-i pi/4)# so

#e^(i (10pi)/4) = e^(ipi/2)# and it's conjugate is #e^(-i pi/2)# and finally

#(1/sqrt2 + i/sqrt2)^10 + (1/sqrt2 - i/sqrt2)^10=e^(ipi/2)+e^(-ipi/2) = i-i=0#