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 that the given expression is of the form \(\cos(\alpha + \beta) \cos \beta + \sin(\alpha + \beta) \sin \beta\), which resembles the cosine addition formula but with a plus sign between the terms.
2. Recall the cosine difference identity: \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Notice that the given expression matches this pattern with \(A = \alpha + \beta\) and \(B = \beta\).
3. Apply the identity by substituting \(A\) and \(B\) into the formula: \(\cos((\alpha + \beta) - \beta) = \cos(\alpha + \beta) \cos \beta + \sin(\alpha + \beta) \sin \beta\).
4. Simplify the argument inside the cosine function: \((\alpha + \beta) - \beta = \alpha\).
5. Write the simplified expression as \(\cos \alpha\), which is a single-term expression.

Key Concepts

Angle Addition Formulas

The angle addition formulas express trigonometric functions of sums of angles in terms of functions of individual angles. For cosine, cos(α + β) = cos α cos β - sin α sin β, and for sine, sin(α + β) = sin α cos β + cos α sin β. These formulas are essential for rewriting and simplifying expressions involving sums of angles.

Trigonometric Identities for Simplification

Trigonometric identities allow the transformation of complex expressions into simpler or more recognizable forms. Recognizing patterns such as products of cosines and sines can help combine terms into a single trigonometric function, facilitating simplification to one term.

Combining Like Terms Using Trigonometric Properties

Combining terms involving sine and cosine functions often requires using properties like distributive law and substitution from known identities. This process helps to rewrite sums of products into a single trigonometric function, which is the goal in simplifying expressions to one term.