Answer:
1/2sin(6x) + 1/2sin(2x)
Explanation:
You can look up the formulas for the product identities for sine and cosine, or you can guess and check using a graphing calculator. I did the calculator solution first (see the first attachment), then looked up the identities so I can tell you what they are (see the second attachment).
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These identities are based on the sum and difference angle identities:
sin(α+β) +sin(α-β) = (sin(α)cos(β) +sin(β)cos(α)) + (sin(α)cos(β) -sin(β)cos(α))
= 2sin(α)cos(β)
Dividing by 2 gives the identity of interest in this problem:
sin(4x)cos(2x) = (1/2)(sin(4x +2x) +sin(4x -2x))
sin(4x)cos(2x) = (1/2)(sin(6x) +sin(2x))