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Product-to-sum identities are a set of trigonometric identities used to simplify trigonometric expressions by converting them from a product of functions to a sum or difference. They can be particularly helpful in solving integrals and simplifying complex expressions in trigonometry.
These identities arise from the angle addition identities and provide a useful alternative representation of products of sine and cosine functions.

• For the cosine product, the identity is:\[\cos(A) \cos(B) = \frac{1}{2} [\cos(A + B) + \cos(A - B)]\]
• The sine and cosine product-to-sum identities include:\[\sin(A) \sin(B) = \frac{1}{2} [\cos(A - B) - \cos(A + B)]\]\[\sin(A) \cos(B) = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\]

By using these identities, we can rewrite products as sums, thus simplifying the expressions significantly, which is often necessary in precalculus exercises.

The cosine function is one of the fundamental trigonometric functions, often denoted as \( \cos(\theta) \), where \( \theta \) is the angle. It represents the x-coordinate of a point on the unit circle.
The cosine function has several key properties:

• Periodicity: The cosine function is periodic with a period of \(2\pi\). This means \( \cos(\theta) = \cos(\theta + 2\pi k) \) for any integer \(k\).
• Symmetry: The cosine function is an even function, meaning \( \cos(-\theta) = \cos(\theta) \). This property proves useful as shown in our solution when simplifying expressions.
• Range: Its range is from -1 to 1, encompassing all possible values that the cosine function can achieve.

Using its symmetry, we find that \( \cos(-5x) = \cos(5x) \), simplifying our final expression in the given exercise.

Precalculus is a branch of mathematics that prepares students for calculus. It covers advanced topics from algebra and trigonometry, offering a solid foundation for differential and integral calculus.
The course often reinforces understanding of trigonometric identities, such as product-to-sum identities, and their importance in simplifying expressions. These identities are crucial tools for students aiming to build their analytical skills.
In precalculus, you may also encounter:

• Analyzing functions, including polynomial, rational, exponential, and logarithmic functions.
• Exploring complex numbers and their properties.
• Understanding sequences, series, and probability.

Mastering these concepts is essential for success in calculus, where learners encounter rates of change and integrals, concepts that often involve expressions simplified by identities like those found in this exercise.