Operations on Complex Numbers in Polar Form

Jan 2013
175
3
Australia
By finding z if z = cis(Ѳ), show that cos (4Ѳ) = cos4(Ѳ) - 6cos2(Ѳ)sin2(Ѳ) + sin4(Ѳ) and that sin(4Ѳ) = 4cos3(Ѳ)sin(Ѳ) - 4cos(Ѳ)sin3(Ѳ).

I know that I first have to use De Moivre's theorem to find z4 when z = cis(Ѳ), which gives me z4 = 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.