In Exercises 69–74, rewrite each expression as a simplified expression containing one term.cos (α + β) cos β + sin (α + β) sin β

Verified step by step guidance

1. Recognize the expression as a form of the angle addition formula for cosine: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).
2. Notice that the given expression \( \cos(\alpha + \beta) \cos \beta + \sin(\alpha + \beta) \sin \beta \) resembles the angle addition formula for cosine, but with a different arrangement.
3. Recall the angle addition formula for cosine: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).
4. Identify that the given expression matches the form of \( \cos(\alpha - \beta) \).
5. Conclude that the expression simplifies to \( \cos(\alpha - \beta) \).

Key Concepts

Angle Addition Formulas

The angle addition formulas are fundamental identities in trigonometry that express the sine and cosine of the sum of two angles. Specifically, they state that cos(α + β) = cos(α)cos(β) - sin(α)sin(β) and sin(α + β) = sin(α)cos(β) + cos(α)sin(β). These formulas are essential for simplifying expressions involving sums of angles.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They include fundamental relationships such as the Pythagorean identity, reciprocal identities, and co-function identities. Understanding these identities is crucial for manipulating and simplifying trigonometric expressions effectively.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a more manageable form, often by combining terms or applying identities. This process can include factoring, using angle addition formulas, or substituting equivalent expressions. Mastery of simplification techniques is vital for solving complex trigonometric problems.