For cos(α - β), which of the following is the correct identity?

The identity for cos(α - β) is derived from the cosine of a difference formula. This states that the cosine of the difference between two angles is equal to the cosine of the first angle multiplied by the cosine of the second angle plus the sine of the first angle multiplied by the sine of the second angle. Therefore, cos(α - β) = cos α cos β + sin α sin β accurately reflects this relationship.

Understanding the components of the identity is crucial. The term cos α cos β represents the product of the cosines of the two angles, which captures how the orientation of these angles affects the resultant cosine value. Similarly, the term sin α sin β accounts for how the sine components combine when the angles are subtracted, thereby preserving the relationships within the trigonometric circle.

This formula fundamentally illustrates how subtracting angles is not merely a straightforward numeric operation but has deep geometrical implications rooted in the properties of the unit circle. The correct identity captures these relationships accurately, making it a fundamental tool in trigonometry for simplifying expressions involving cosine of angles.