bentuk sederhana dari (sin 4x-sin x)^2 + (cos 4x - cosx)^2

[tex]
\left(\sin{(4x)}-\sin{(x)}\right)^2+\left(\cos{(4x)}-\cos{(x)}\right)^2\\
=\sin^2{(4x)}-2\sin{(4x)}\sin{(x)}+\sin^2{(x)}+\cos^2{(4x)}-2\cos{(4x)}\cos{(x)}+\cos^2{(x)}\\
=\left(\sin^2{(4x)}+\cos^2{(4x)}\right)+\left(\sin^2{(x)}+\cos^2{(x)}\right)-2\sin{(4x)}\sin{(x)}-2\cos{(4x)}\cos{(x)}\\
=1+1-2\left(\cos{(4x)}\cos{(x)}+\sin{(4x)}\sin{(x)}\right)\\
=2-2\cos{(4x-x)}\\
=2-2\cos{(3x)}
[/tex]

(sin 4x - sin x)² + (cos 4x - cos x)²
= (sin² 4x + sin² x - 2sin 4x sin x) + (cos² 4x + cos² x - 2cos 4x cos x)
= (sin² 4x + cos² 4x) + (sin² x + cos² x) - 2(cos 4x cos x + sin 4x sin x)
= 1 + 1 - 2 cos (4x - x)
= 2(1 - cos 3x)