Prove that cos² 2x – cos² 6x = sin 4x sin 8x

To prove cos² 2x – cos² 6x = sin 4x sin 8x

RHS = sin 4x sin 8x

LHS = cos² 2x – cos² 6x

= (1 – sin² 2x) – (1 – sin² 6x)

= 1 – sin² 2x – 1 + sin² 6x

= sin² 6x – sin² 2x

= (sin 6x – sin 2x) (sin 6x + sin 2x) [a² – b² = (a – b)(a + b)]

Rearranging we get

LHS = (2 × sin 2x × cos 2x)(2 × sin 4x × cos 4x)

LHS = sin 4x sin 8x [∵ 2sin A cos A = sin 2A]

∴ LHS = RHS

Hence, proved.