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 75° ﹣ cos 15°

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1. Identify the trigonometric identity for the difference of cosines: \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \).
2. Substitute \( A = 75^\circ \) and \( B = 15^\circ \) into the identity.
3. Calculate \( \frac{A+B}{2} = \frac{75^\circ + 15^\circ}{2} = 45^\circ \).
4. Calculate \( \frac{A-B}{2} = \frac{75^\circ - 15^\circ}{2} = 30^\circ \).
5. Express the difference as a product: \( -2 \sin(45^\circ) \sin(30^\circ) \).

Key Concepts

Sum and Difference Formulas

The sum and difference formulas in trigonometry allow us to express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of those angles. For example, the cosine difference formula states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B). These formulas are essential for simplifying expressions involving trigonometric functions.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values allows for quick calculations and simplifications in trigonometric problems. For instance, cos(30°) = √3/2 and sin(30°) = 1/2 are examples of exact values that can be used in various calculations.

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences, which can simplify calculations. For example, the formula cos(A)cos(B) = 1/2[cos(A + B) + cos(A - B)] is useful for transforming expressions like cos(75°) - cos(15°) into a more manageable form. This transformation is particularly helpful when finding exact values or further simplifying trigonometric expressions.