Use the product-to-sum identities to rewrite the following expression as a sum or difference.3cos(2x)cos(4x)
Question
Use the product-to-sum identities to rewrite the following expression as a sum or difference.3cos(2x)cos(4x)
Solution
The product-to-sum identities are:
cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)] sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)] sin(A)cos(B) = 1/2[sin(A+B) + sin(A-B)]
We can use the first identity to rewrite the expression 3cos(2x)cos(4x) as a sum or difference.
Let A = 2x and B = 4x.
Then, cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)] becomes:
3cos(2x)cos(4x) = 3/2[cos(2x-4x) + cos(2x+4x)] = 3/2[cos(-2x) + cos(6x)] = 3/2[cos(2x) + cos(6x)]
So, 3cos(2x)cos(4x) can be rewritten as 3/2[cos(2x) + cos(6x)].
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