By DeMoivre's theorem,
[tex]\cos(4\theta)+i\sin(4\theta)=(\cos\theta+i\sin\theta)^4[/tex]
Expanding the right side gives
[tex]\cos^4\theta+4i\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta-4i\cos\theta\sin^3\theta+\sin^4\theta[/tex]
Equating imaginary parts tells us
[tex]\sin(4\theta)=4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta[/tex]
(Not sure what you mean by sin(e) and cos(e)...)
