In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x

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1. Recall the product-to-sum identity for the product of sines: \(\sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)]\).
2. Identify the angles in the problem: here, \(A = 6x\) and \(B = 4x\).
3. Substitute \(A\) and \(B\) into the identity: \(\sin 6x \sin 4x = \frac{1}{2} [\cos (6x - 4x) - \cos (6x + 4x)]\).
4. Simplify the expressions inside the cosine functions: \(\cos (6x - 4x) = \cos 2x\) and \(\cos (6x + 4x) = \cos 10x\).
5. Write the final expression as a sum or difference: \(\sin 6x \sin 4x = \frac{1}{2} [\cos 2x - \cos 10x]\).

Key Concepts

Product-to-Sum Identities

Product-to-sum identities transform products of sine and cosine functions into sums or differences of trigonometric functions. For example, the product sin A sin B can be expressed as a difference of cosines: (1/2)[cos(A - B) - cos(A + B)]. This simplifies integration and solving trigonometric equations.

Trigonometric Angle Notation and Manipulation

Understanding how to handle angles in trigonometric expressions is essential. Here, angles like 6x and 4x represent multiples of a variable, and correctly applying operations like addition and subtraction on these angles is crucial when using identities.

Simplification of Trigonometric Expressions

After applying identities, simplifying the resulting expressions by combining like terms or factoring is important. This step ensures the final answer is in its simplest sum or difference form, making it easier to interpret or use in further calculations.