Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value.cos 4x + cos 2x

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1. Identify the trigonometric identity for the sum of cosines: \( \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).
2. Assign \( A = 4x \) and \( B = 2x \) to the identity.
3. Substitute \( A \) and \( B \) into the identity: \( \cos 4x + \cos 2x = 2 \cos \left( \frac{4x + 2x}{2} \right) \cos \left( \frac{4x - 2x}{2} \right) \).
4. Simplify the expressions inside the cosines: \( \frac{4x + 2x}{2} = 3x \) and \( \frac{4x - 2x}{2} = x \).
5. Express the sum as a product: \( 2 \cos 3x \cos x \).

Key Concepts

Sum-to-Product Formulas

Sum-to-product formulas are trigonometric identities that express sums or differences of sine and cosine functions as products. For example, the formula for the sum of cosines states that cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2). These formulas are essential for simplifying expressions like cos 4x + cos 2x into a product form, which can then be evaluated more easily.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They include fundamental identities such as the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for manipulating trigonometric expressions and solving equations, as they provide the foundational relationships between different trigonometric functions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent at key angles, such as 0°, 30°, 45°, 60°, and 90°. These values are often expressed as fractions or radicals and are essential for evaluating trigonometric expressions without a calculator. Knowing these exact values allows for precise calculations when simplifying expressions derived from trigonometric identities.