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

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1. Recognize the given expression: \( \sin(\alpha - \beta) \cos \beta + \cos(\alpha - \beta) \sin \beta \).
2. Identify that this expression resembles the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
3. Notice that the expression matches the form of \( \sin(A + B) \) where \( A = \alpha - \beta \) and \( B = \beta \).
4. Apply the sine addition formula: \( \sin((\alpha - \beta) + \beta) = \sin \alpha \).
5. Conclude that the simplified expression is \( \sin \alpha \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle identities. Understanding these identities is crucial for simplifying trigonometric expressions, as they provide the foundational relationships between different trigonometric functions.

Angle Sum and Difference Formulas

The angle sum and difference formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β). These formulas are essential for rewriting expressions involving the sine and cosine of combined angles, which is necessary for simplifying the given expression.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting complex expressions into a more manageable form, often with fewer terms. This process typically utilizes trigonometric identities to combine or reduce terms. Mastery of simplification techniques is vital for solving trigonometric problems efficiently, as it allows for clearer analysis and easier computation.