Prove that : sin 2x + 2 sin 4x + sin 6x = 4 cos2x sin 4x

Solution:

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

= (sin 6x + sin 2x) + 2 sin 4x …(i)

We know sin A + sin B = 2 sin (A+B)/2 cos (A-B)/2

(i) becomes 2 sin 4x cos 2x + 2 sin 4x 

= 2 sin 4x (cos 2x + 1)

= 2 sin 4x (2 cos2x – 1 + 1)

= 2 sin 4x (2 cos2x)

= 4 cos2x sin 4x

= RHS

Hence proved.

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