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Trigonometric identities play a crucial role in simplifying expressions and solving problems involving angles and triangles in precalculus. One of the most used identities in trigonometry is the product-to-sum identity, which provides a method to rewrite the product of two cosine functions as the sum or difference of two cosines. This is particularly useful in problems where products appear, but a sum or difference is more manageable.

One should recall that a product-to-sum identity can be stated as \(\cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B))\). Such identities help in converting difficult products into simpler sum or difference terms that are easier to work with. They are derived from the sum and difference formulas of cosine function, wedging their importance in the field of trigonometry.

The cosine function, one of the primary trigonometric functions, is widely used to determine the horizontal distance a point on a unit circle has from the origin. In the expression \(\cos 13x \cos -5x\), the cosine function is initially presented in a product format. Understanding how the cosine function behaves is essential to applying identities correctly.

To dive deeper, the cosine function represents the adjacent side over the hypotenuse when considering a right-angled triangle. However, in a broader sense, the function is applicable in cycles and waves and is especially useful in modeling periodic phenomena. Recognizing the cosine's properties, such as its evenness (\(\cos A = \cos (-A)\)), is beneficial when deconstructing products of cosines into more workable forms.

Precalculus is a course that prepares students for calculus, covering various topics that include algebraic, trigonometric, and exponential functions. An understanding of trigonometric identities, including the product-to-sum identity, is a foundational aspect of precalculus. It helps students to navigate through complex trigonometric expressions and move towards solving calculus problems.

Students learn to handle problems involving angles, triangles, and periodic functions, gearing up with the necessary skills for calculus. Simplifying expressions using trigonometric identities, as shown in the exercise above, is a practical application of precalculus. It's a staircase to higher mathematics, where such manipulations become a toolset to tackle limits, derivatives, and integrals.