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Trigonometric identities are equations that express a connection between the trigonometric functions and are true for all values of the variables involved. They are incredibly useful for simplifying expressions, solving trigonometry problems, and for computational purposes.

One of the most widely used identities is the product-to-sum formulas, which convert the product of two sine or cosine functions into a sum or difference of two trigonometric functions. For example, \(\cos a \sin b = \frac{1}{2} [\sin(a+b) - \sin(a-b)]\). These identities are especially helpful when integrating products of trigonometric functions or when finding the Fourier series of a function.

Understanding and being able to apply trigonometric identities can greatly enhance the ability to manipulate and solve trigonometry problems with ease.

The sine function is a fundamental trigonometric function and has several important properties which make it unique and very useful in trigonometry problem solving. The sine function, denoted as \(\sin\), is periodic with a period of \(2\pi\), meaning that \(\sin (x + 2\pi) = \sin x\) for all values of \(x\). This periodicity allows us to find the sine of angles greater than \(2\pi\) by reducing them into an equivalent angle within the interval \(0, 2\pi\) range.

The function is also an odd function, which means that \(\sin(-x) = -\sin(x)\), reflecting the property of sine across the origin. Furthermore, the sine function has a known value at specific familiar angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi,\ldots\). For instance, \(\sin \pi = 0\) and \(\sin \frac{\pi}{2} = 1\). These properties can significantly simplify trigonometry problems when correctly applied.

Problem solving in trigonometry often involves the application of trigonometric identities and the properties of trigonometric functions to simplify expressions and solve equations. The key to solving trigonometry problems is to be familiar with the various identities and properties, such as the Pythagorean identities, angle sum and difference identities, and double and half-angle formulas.

In the context of the given exercise, applying the product-to-sum formula, which is a type of trigonometric identity, reduces the problem to a simpler form that can be easily evaluated. This systematic approach involves identifying relevant values or angles and methodically applying the correct formulas. By breaking down complex trigonometric expressions into simpler forms, we enhance our chances of solving the problems correctly.

Always remember to check whether the expression can be simplified further by using known sine and cosine values or by using properties like periodicity and the even-odd characteristics of the trigonometric functions.

Angles addition and subtraction form the foundation of many trigonometric identities, including the product-to-sum formulas. In trigonometry, the sine and cosine of an angle sum or difference can be expressed in terms of the sines and cosines of the individual angles. This is extremely useful when dealing with trigonometric functions of composite angles.

For instance, given \(\sin(a + b)\) and \(\sin(a - b)\), we can express these as functions of \(\sin a\), \(\sin b\), \(\cos a\), and \(\cos b\). The sine of an angle sum is \(\sin(a + b) = \sin a \cos b + \cos a \sin b\), and for the angle difference, \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). Note that this concept is crucial for the step by step solution given in the exercise, as it allows us to transform the product of trigonometric functions into a sum. By mastering angle addition and subtraction, we unlock the ability to simplify many trigonometric expressions and thus solve complex problems more effectively.