91Ó°ÊÓ

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities simplify the process of integrating complex trigonometric expressions. For instance, the product-to-sum identities, like the one used in this exercise, allow us to convert a product of trigonometric functions into a sum or difference of functions.

Some common trigonometric identities include:

• Pythagorean identities: \( \sin^2 \theta + \cos^2 \theta = 1\)
• Angle sum identities: \( \sin(A+B) = \sin A \cos B + \cos A \sin B\)
• Product-to-sum identity as in the exercise: \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]\)

Recognizing which identity to use can make integration problems much simpler and more manageable.

Integration techniques are methods used to find the integral, or antiderivative, of a function. In the context of trigonometric functions, these techniques often involve algebraic manipulation followed by integration of simpler terms.

Here's a general approach for using integration techniques:

• Identify any applicable trigonometric or algebraic identities.
• Rewrite the expression using these identities to simplify the form.
• Integrate the expression after simplification, often using basic integration formulas.

For example, when integrating a product of trigonometric functions, applying a trigonometric identity to break it down into simpler parts allows us to use straightforward integration techniques like recognizing patterns from basic integrals such as \( \int \cos x \, dx = \sin x + C\).
Recognizing the correct identity and technique is crucial to obtaining the solution efficiently.

Product-to-sum identities are particularly useful in simplifying the integration of products of trigonometric functions. These identities convert a product of two trig functions into a sum or difference of trigonometric functions.

In this exercise, we used the identity:

• \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \)

By substituting \( A = s \) and \( B = 2s \), this identity helps rewrite \( \sin s \sin 2s \) as a simpler expression: \( \frac{1}{2} [\cos(-s) - \cos(3s)] \.\)

This simplification turns the integration problem into a task that involves simpler trigonometric functions, which are easier to integrate. Once in a simplified form, we can apply basic integration formulas to solve the problem. Product-to-sum identities not only simplify the integration process but also enhance understanding of the relationship between trigonometric functions.