Final answer:
To write the sum cos(4x) - cos(8x) as a product, we use the sum-to-product formula, resulting in 2 cos(2x) cos(6x).
Step-by-step explanation:
The expression cos(4x) - cos(8x) can be written as a product using trigonometric identities. Specifically, we will use the sum-to-product formulas to transform the sum of cosines into a product of sine and cosine. Using the identity cos a + cos b = 2 cos((a + b)/2) cos((a - b)/2), where a = 4x and b = -8x, we get:
- cos(4x) + cos(-8x) = 2 cos((4x - 8x)/2) cos((4x + 8x)/2)
- cos(4x) - cos(8x) = 2 cos(-2x) cos(6x)
- Since cos(-θ) = cos(θ), this simplifies to: 2 cos(2x) cos(6x)
To write the sum as a product of cos(4x) - cos(8x), we can use the trigonometric identity:
cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)
So, we can rewrite the expression as:
cos(4x) - cos(8x) = -2sin((4x + 8x)/2)sin((4x - 8x)/2)
Simplifying further:
cos(4x) - cos(8x) = -2sin(6x)sin(-2x)
Therefore, the sum cos(4x) - cos(8x) can be written as the product 2 cos(2x) cos(6x).