How do you use DeMoivre's Theorem to simplify $(cos(pi/4)+isin(pi/4))^12$ ?

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Answer 1 · 2016-08-12T17:22:56

$-1$

De Moivre's identity says

$e^{i x} = cos(x) + i sin(x)$

Now, taking

$e ^{i pi/4}= cos(pi/4)+isin(pi/4)$ we have

$(cos(pi/4)+isin(pi/4))^12 = e^{i(12pi/4)} = e^{i3pi} = e^{i(pi+2pi)}$

and finally

$(cos(pi/4)+isin(pi/4))^12=e^{ipi} = cos(pi)+isin(pi) = -1$

How do you use DeMoivre's Theorem to simplify (cos(pi/4)+isin(pi/4))^12(cos(pi/4)+isin(pi/4))^12?