91Ó°ÊÓ

The sum-to-product identities are vital tools in trigonometry. They transform a sum or difference of trigonometric functions into a product, which is often easier to work with. For instance, when you have expressions like \(\cos A + \cos B\), the sum-to-product identity allows you to rewrite it as \(2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\).
This conversion can significantly simplify calculations by reducing complex trigonometric sums into more manageable products. In the given exercise, applying this identity to \(\cos 45^{\circ} + \cos 15^{\circ}\) enabled us to consolidate the expression into a non-trivial product form, facilitating further simplification.

Simplifying angles is a crucial step in applying sum-to-product identities effectively. After substituting angles \(A = 45^{\circ}\) and \(B = 15^{\circ}\) into the identity, the next task is computing their average and difference:

• The average is \(\frac{45^{\circ} + 15^{\circ}}{2} = 30^{\circ}\).
• The difference calculates as \(\frac{45^{\circ} - 15^{\circ}}{2} = 15^{\circ}\).

Breaking down these angles ensures a straightforward pathway to utilize known trigonometric values, such as those for \(30^{\circ}\) and \(15^{\circ}\). Keeping these simplified forms in check keeps your calculations organized and approachable as we proceed to solve the expression.

The cosine function is a fundamental trigonometric function, representing the adjacent side's ratio to the hypotenuse in a right triangle. Knowing the precise values for certain angles can simplify matters immensely.
Particularly, the cosine of \(30^{\circ}\) is \(\frac{\sqrt{3}}{2}\), a frequently used trigonometric value. By substituting this known value, you can simplify expressions effectively. In our problem, the previously converted product term, \(2 \cos 30^{\circ} \cos 15^{\circ}\), was simplified using \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\). This resulted in a cleaner and reduced expression \(\sqrt{3} \cos 15^{\circ}\).
Understanding the fundamental values of the cosine function is crucial for solving trigonometric expressions and deeply understanding their underlying principles.