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Question
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Solution
L.H.S. = sin 2x + 2 sin 4x + sin 6x
= [sin 2x + sin 6x] + 2 sin 4x
= `2sin ((6x + 2x)/2)cos ((6x - 2x)/2) + 2sin 4x`
`[∵ sin A + B = 2 sin ((A+ B)/2) cos ((A - B)/2)]`
= 2 sin 4x cos (– 2x) + 2 sin 4x
= 2 sin 4x cos 2x + 2 sin 4x
= 2 sin 4x (cos 2x + 1)
= 2 sin 4x (2 cos2 x – 1 + 1)
= 2 sin 4x (2 cos2 x)
= 4cos2 x sin 4x
= R.H.S.
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