Which of the following represents the cosine double-angle identity?

The cosine double-angle identity is used to express the cosine of an angle that is double another angle, specifically in the form of cos(2θ). The correct representation of this identity is found in the option stating that cos(2θ) can be expressed as 2cos²(θ) - 1.

To understand why this is correct, consider the fundamental relationships in trigonometry. The double-angle identities derive from the addition formulas for cosine. Specifically, using the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b), if we let both a and b equal θ, we arrive at:

cos(2θ) = cos(θ + θ) = cos(θ)cos(θ) - sin(θ)sin(θ) = cos²(θ) - sin²(θ).

This expression can be further manipulated utilizing the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1. By expressing sin²(θ) in terms of cos²(θ), we can substitute sin²(θ) with (1 - cos²(θ)), leading to:

cos(2θ) = cos²(θ) -