Write each expression as a sum or difference of trigonometric functions. See Example 7.
8 sin 7x sin 9x

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Recall the product-to-sum identity for sine functions: $\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$.

Identify $A$ and $B$ in the given expression: here, $A = 7x$ and $B = 9x$.

Apply the identity to rewrite $\sin 7x \sin 9x$ as $\frac{1}{2} [\cos(7x - 9x) - \cos(7x + 9x)]$.

Simplify the arguments inside the cosine functions: \$7x - 9x = -2x$ and $7x + 9x = 16x$.

Multiply the entire expression by 8 (from the original expression $8 \sin 7x \sin 9x$) to get $8 \times \frac{1}{2} [\cos(-2x) - \cos(16x)]$, which simplifies to $4 [\cos(-2x) - \cos(16x)]$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences, simplifying expressions and integrals. For example, the formula for the product of two sine functions is sin A sin B = 1/2 [cos(A - B) - cos(A + B)]. This is essential for rewriting 8 sin 7x sin 9x as a sum or difference.

Trigonometric Function Properties

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

Algebraic Manipulation of Trigonometric Expressions

Algebraic skills are necessary to factor constants and apply identities correctly. For instance, factoring out the constant 8 and applying the product-to-sum formula requires careful distribution and simplification to express the product as a sum or difference.

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