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The product-to-sum formulas are valuable tools in trigonometry. They help convert the product or difference of trigonometric functions into a sum or difference of trigonometric terms. This is particularly useful in simplifying complex expressions and solving trigonometric equations. The specific product-to-sum formula used in the exercise is for the difference of cosines:
\[\text{cos A - cos B = -2 sin \frac{(A + B)}{2} sin \frac{(A - B)}{2}} \]
In our example, we look at \(\text{cos 4x - cos 2x}\). By applying the product-to-sum formula, we first identify our A and B values, which are 4x and 2x, respectively. Substituting these values into the formula transforms the difference of cosines into a product of sines.
Understanding these conversions can make working with trigonometric functions much more manageable, especially when dealing with integration, differentiation, or equation solving.

Simplification in mathematics means making an expression easier to work with or understand. In trigonometry, simplification involves converting complex trig expressions into simpler forms.
In the exercise, we start with \(\text{cos 4x - cos 2x}\). After applying the product-to-sum formula, we get:
\[\text{-2 sin \frac{(4x + 2x)}{2} sin \frac{(4x - 2x)}{2}} \]
Simplifying inside the sine functions:
\[ \text{-2 sin (3x) sin (x)} \]
The goal is to make the expression more straightforward or easier to understand. Simplification helps in solving equations and making calculations less cumbersome. It also plays a crucial role in calculus and other advanced mathematical topics.

Trigonometric identities are equations that hold true for all values within the functions' domains. They are foundational tools used to simplify, solve, and transform trigonometric expressions.
Some common identities include:

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• Pythagorean identities
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• Angle sum and difference identities
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• Double-angle identities
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• Half-angle identities
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• Product-to-sum and sum-to-product identities

In the exercise, the product-to-sum identity for cosine is used to transform: \( \text{cos 4x - cos 2x} \) into a product of sines.
This approach leverages the trigonometric identities to simplify the expression into \( \text{-2 sin (3x) sin (x)} \). Knowing these identities allows for more straightforward manipulation and understanding of trigonometric expressions, aiding in solving trigonometric equations or conducting further mathematical analysis.