Use the Binomial Expansion.
By finding z if z = cis(Ѳ), show that cos (4Ѳ) = cos^{4}(Ѳ) - 6cos^{2}(Ѳ)sin^{2}(Ѳ) + sin^{4}(Ѳ) and that sin(4Ѳ) = 4cos^{3}(Ѳ)sin(Ѳ) - 4cos(Ѳ)sin^{3}(Ѳ).
I know that I first have to use De Moivre's theorem to find z^{4} when z = cis(Ѳ), which gives me z^{4} = cis(4Ѳ) = cos(4Ѳ) + sin(4Ѳ)i.
I also know that I need to expand [cos(Ѳ) + isin(Ѳ)]^{4}, but that's where I get stuck. What's the best way to go about expanding this expression?
Thanks in advance.