Question

How do you use the sum to product formulas to write the sum or difference `cos6x+cos2x` as a product?

Answer 1

`cos(6x)+cos(2x)=2cos4xcos2x`

Explanation:

`cos(A+B) + cos(A-B) = 2cosAcosB`
Thus by comparing the left side of the above equation with the one we are given, we see that `A+B=6x` and `A-B=2x`

Solving simultaneously, we see that `A=4x` and `B=2x`

Thus, `cos(6x)+cos(2x)=2cos4xcos2x`

PROOF OF THIS SUM TO PRODUCT:
`cos(A+B) + cos(A-B) = cosAcosB - sinAsinB + cosAcosB + sinAsinB`
`=2cosAcosB`