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The product-to-sum formulas are essential tools in trigonometry that allow us to express the product of trigonometric functions as a sum or difference. This can be especially useful in simplifying complex expressions. In our example, the product \(3 \cos 4x \cos 7x\) can be transformed into a sum using these formulas. The particular formula used here is:

• \( \cos A \cos B = \frac{1}{2}\left(\cos(A + B) + \cos(A - B)\right) \)

By applying this formula, the expression \(\cos 4x \cos 7x\) is rewritten as:

• \( \frac{1}{2} (\cos(11x) + \cos(3x)) \)

This finding simplifies the task of further operations on the expression, as it reduces a product involving angles into a manageable sum. Keeping these formulas in mind is crucial when dealing with trigonometric identities.

Simplification involves transforming a trigonometric expression into a more manageable form without changing its fundamental value. The problem provided asks us to transform a product expression into a sum, which ultimately results in a much easier format to work with. The process begins with recognizing the trigonometric identity that fits the expression you are working with.

In our example, begin by applying the product-to-sum identity to \(\cos 4x \cos 7x\). The result is an expression already simplified to a sum of cosines of different angles:

• \( \frac{1}{2} (\cos(11x) + \cos(3x)) \)

Finally, account for any coefficients initially present in the equation, in this case, the coefficient "3" is multiplied by the simplified sum to maintain equivalence. The ending expression:

• \( \frac{3}{2} \cos(11x) + \frac{3}{2} \cos(3x) \)

enables further algebraic manipulations or integrations to be done with less complexity.

The cosine function, denoted as \(\cos\), is a fundamental part of trigonometry and is crucial in simplifying expressions like the one given in the exercise. It's a periodic function known for its ability to describe oscillations, such as waves.

An important property to note is the even nature of the cosine function, meaning that \(\cos(-x) = \cos(x)\). This characteristic was particularly useful in our solution where \(\cos(11x)\) and \(\cos(3x)\) were considered.

• Changing signs inside a cosine function does not affect its value.
• This intrinsic property simplifies interpretation and manipulation of expressions.

In trigonometric identities and simplifications, understanding properties like these aids greatly in transforming and maneuvering mathematical expressions to the desired format.