$\sin3x=\sin(2x+x)=\sin2x\cos x+\sin x\cos2x$
$=2\sin x\cos x.\cos x+\sin x(1-2\sin^2x)$
$=2\sin x.(1-\sin^2x)+\sin x(1-2\sin^2x)$
$=2\sin x-2\sin^3x+\sin x-2\sin^3x$
$=3\sin x-4\sin^3x$
$\cos3x=\cos(x+2x)=\cos x\cos2x-\sin x\sin2x$
$=\cos x(2\cos^2x-1)-2\sin^2x\cos x$
$=\cos x(2\cos^2x-1)-2\cos x(1-\cos^2x)$
$=2\cos^3x-\cos x-2\cos x+2\cos^3x$
$=4\cos^3x-3\cos x$