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Prove that sin2x + 2 sin4x + sin6x = 4 cos2 x sin4x

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sin 2x= sin (4x-2x) = sin 4x cos 2x - cos 4x sin 2x 
sin 4x = sin 4x 
sin 6x = sin (4x+2x) = sin 4x cos 2x + cos 4x sin 2x 

Adding we get:

= sin 2x + 2 sin 4x + sin 6x 

= 2 sin 4x cos 2x + 2sin 4x 

= 2 sin 4x( cos 2x + 1) 
= 2 sin 4x ( 2 cos2 x) 

= 4 sin 4x cos2 x

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