Using de Moivre's identity
e^(ix)=cos(x)+isin(x)eix=cos(x)+isin(x) we obtain
e^(4ix) = cos(4x)+isin(4x) = (cos(x)+isin(x))^4e4ix=cos(4x)+isin(4x)=(cos(x)+isin(x))4
but
(cos(x)+isin(x))^4=Cos(x)^4 - 6 Cos(x)^2 Sin(x)^2 + Sin(x)^4 +
i (4 Cos(x)^3 Sin(x) - 4 Cos(x) Sin(x)^3)(cos(x)+isin(x))4=cos(x)4−6cos(x)2sin(x)2+sin(x)4+i(4cos(x)3sin(x)−4cos(x)sin(x)3)
Keeping the real component
cos(4x)=Cos(x)^4 - 6 Cos(x)^2 Sin(x)^2 + Sin(x)^4 =cos(4x)=cos(x)4−6cos(x)2sin(x)2+sin(x)4=
=Cos(x)^4+ 6 Cos(x)^2(1-cos(x)^2)+(1-cos(x)^2)^2==cos(x)4+6cos(x)2(1−cos(x)2)+(1−cos(x)2)2=
=8 cos(x)^4 - 8 cos(x)^2 +1=8cos(x)4−8cos(x)2+1
