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

Verified step by step guidance

1. Recognize that the given expression is of the form \(\sin(A) \cos(B) + \cos(A) \sin(B)\), where \(A = \alpha - \beta\) and \(B = \beta\).
2. Recall the sine addition formula: \(\sin(X + Y) = \sin X \cos Y + \cos X \sin Y\).
3. Apply the sine addition formula to the expression \(\sin(\alpha - \beta) \cos \beta + \cos(\alpha - \beta) \sin \beta\), which matches the pattern \(\sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B)\).
4. Substitute back the values of \(A\) and \(B\) to get \(\sin((\alpha - \beta) + \beta)\).
5. Simplify the argument inside the sine function: \((\alpha - \beta) + \beta = \alpha\), so the expression simplifies to \(\sin \alpha\).

Key Concepts

Angle Sum and Difference Identities

These identities express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, sin(α - β) = sin α cos β - cos α sin β. Recognizing and applying these identities helps simplify complex expressions involving multiple angles.

Trigonometric Function Properties

Understanding the basic properties and relationships of sine and cosine functions, such as their periodicity and symmetry, is essential. This knowledge aids in manipulating and combining terms to achieve simpler or more recognizable forms.

Expression Simplification Techniques

Simplifying trigonometric expressions often involves factoring, combining like terms, and substituting identities. Mastery of these algebraic techniques allows one to rewrite expressions as a single trigonometric term, making them easier to interpret or use in further calculations.