If you have some knowledge of complex numbers: This follows pretty much immediately from Euler's formula and DeMoivre's theorem,
[tex]e^{ix}=\cos x+i\sin x[/tex] (Euler)
[tex](\cos x+i\sin x)^n=\cos nx+i\sin nx[/tex] (DeMoivre)
so that [tex]\sin 4x[/tex] is the imaginary part of the expanded left hand side.
If that's unfamiliar to you, you can make use of several identities to expand [tex]\sin4x[/tex]:
[tex]\sin4x=2\sin2x\cos2x[/tex] (double angle sine)
[tex]=4\sin x\cos x(\cos^2x-\sin^2x)[/tex] (double angle sine and cosine)
[tex]=4\sin x\cos^3x-4\sin^3x\cos x[/tex]
[tex]=4\sin x\cos x\cos^2x-4\sin^3x\cos x[/tex]
[tex]=4\sin x\cos x(1-\sin^2x)-4\sin^3x\cos x[/tex] (Pythagorean)
[tex]=4\sin x\cos x-4\sin^3x\cos x-4\sin^3x\cos x[/tex]
[tex]=4\sin x\cos x-8\sin^3x\cos x[/tex]
