To prove sin 2x + 2 sin 4x + sin 6x = 4 cos²x sin 4x

RHS = 4 cos² x sin 4x

LHS = sin 2x + 2 sin 4x + sin 6x

= 2 sin 4x + sin 6x + sin 2x

We know,

LHS = 2 sin 4x + 2 sin 4x cos 2x

= 2 sin 4x × (1 + cos 2x)

= 2 sin 4x × 2 cos² x [∵ cos2x = 2cos² x – 1]

LHS = 4 cos² x sin 4x

∴ LHS = RHS

Hence, proved.