Final answer:
The expression cos6x - cos2x can be written as the product of trigonometric functions using the cosine sum-to-product identity, resulting in -2 cos(4x) cos(2x).
Step-by-step explanation:
To express cos6x - cos2x as a product of trigonometric functions, we can use the cosine sum-to-product identity which comes from the sum and difference formulas for cosine. The formula is as follows:
cosα + cosβ = 2 cos((α + β)/2) cos((α - β)/2)
Applying it to our expression with α = 6x and β = 2x, we get:
cos6x - cos2x = -(- cos6x + cos2x)
Now using the sum-to-product formula:
cos6x - cos2x = -2 cos((6x + 2x)/2) cos((6x - 2x)/2)
cos6x - cos2x = -2 cos(4x) cos(2x)
Thus, the given expression can be written as -2 cos(4x) cos(2x).