De Moivre's theorem states that for any real number θ and integer n it holds that
(cos(θ)+isin(θ))n=cos(nθ)+isin(nθ).Let n = 3, then
(cos(θ)+isin(θ))3=cos(3θ)+isin(3θ).
Then use the Binomial theorem to expand the terms in the brackets:
cos3(θ)+3isin(θ)cos2(θ)−3sin2(θ)cos(θ)−isin3(θ)=cos(3θ)+isin(3θ).
Thus, equating the real and imaginary parts, we get
cos3(θ)−3sin2(θ)cos(θ)=cos(3θ),3sin(θ)cos2(θ)−sin3(θ)=sin(3θ).Then
cos(3θ)=cos3(θ)−3(1−cos2(θ))cos(θ)=4cos3(θ)−3cos(θ).
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