Write each expression as a sum or difference of trigonometric functions. See Example 7.
5 cos 3x cos 2x

Verified step by step guidance

1. Recall the product-to-sum identity for cosine functions: \(\cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)]\).
2. Identify the angles in the expression: here, \(A = 3x\) and \(B = 2x\).
3. Apply the product-to-sum formula to \(\cos 3x \cos 2x\): \(\cos 3x \cos 2x = \frac{1}{2} [\cos(3x + 2x) + \cos(3x - 2x)]\).
4. Simplify the angles inside the cosine functions: \(\cos(3x + 2x) = \cos 5x\) and \(\cos(3x - 2x) = \cos x\).
5. Multiply the entire expression by 5 (from the original expression \(5 \cos 3x \cos 2x\)) to get the sum or difference form: \(5 \cos 3x \cos 2x = 5 \times \frac{1}{2} [\cos 5x + \cos x]\).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences, simplifying expressions. For example, the formula for cos A cos B is (1/2)[cos(A+B) + cos(A−B)]. These formulas are essential for rewriting products like 5 cos 3x cos 2x as sums or differences.

Trigonometric Function Properties

Understanding the properties of cosine and sine functions, such as periodicity and symmetry, helps in manipulating and simplifying expressions. Recognizing how angles combine in sums or differences is crucial when applying product-to-sum identities.

Algebraic Manipulation of Trigonometric Expressions

Skill in algebraic manipulation, including factoring constants and rearranging terms, is necessary to correctly apply identities and rewrite expressions. For instance, factoring out the constant 5 before applying product-to-sum formulas ensures clarity and accuracy.