Problem 61 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\sin x \sin y=\frac{1}{2}[\cos (x-y)-\cos (x+y)]$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity $$\sin x \sin y = \frac{1}{2}[\cos (x - y) - \cos (x + y)]$$ using the angle subtraction and angle addition formulae for cosine.

Step 1: Recall the angle addition and subtraction formulae for cosine

Before proceeding, we need to recall the following trigonometric identities: $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$ $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$

Step 2: Write the half difference of the two cosine identities

Now, we'll subtract the second identity from the first identity and then divide the result by 2 to obtain an expression for the difference of the two cosines. $$\frac{\cos(A - B) - \cos(A + B)}{2} = \frac{(\cos A \cos B + \sin A \sin B) - (\cos A \cos B - \sin A \sin B)}{2}$$

Step 3: Simplify the expression

Simplify the expression on the right-hand side and attempt to match the given identity: $$\frac{2\sin A \sin B}{2} = \sin A \sin B$$

Step 4: Relate the expression to the given identity

Now relate this expression to the given identity by substituting x and y for A and B respectively: $$\sin x \sin y = \frac{1}{2}[\cos (x - y) - \cos (x + y)]$$ As we can see, we have proved the given identity using the angle subtraction and angle addition formulae for cosine.

Key Concepts

Angle Addition Formula

The Angle Addition Formula for cosine is a powerful tool in trigonometry that helps expand expressions involving angles. Specifically, when you have two angles, \( A \) and \( B \), the formula for their cosine sum is:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

This formula is particularly useful because it breaks down the cosine of a sum of two angles into products of sines and cosines.
This allows us to calculate the cosine of more complex angles by using the values of simpler ones.
In exercises like proving identities, these formulae are crucial, as they help transition from complex expressions into simpler or more familiar forms that are easier to compare or simplify. Essentially, understanding and applying the Angle Addition Formula is a fundamental skill in advancing in trigonometry.

Angle Subtraction Formula

Closely related to the Angle Addition Formula is the Angle Subtraction Formula. This formula is used for cosine differences:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

Here, the expression breaks down a cosine difference into a mixture of products of sine and cosine functions.
Such a formula is key when working on transformations and manipulations in trigonometric identities.
In our original exercise, the subtraction and addition formulas are employed to simplify and rearrange expressions.
When subtracting two cosine-related expressions, as was done in the solution, this formula aids in progressively adapting the terms.
Ultimately, such formulas facilitate easier management of trigonometric equations by transforming them into recognizable and solvable forms.

Trigonometric Proofs

Trigonometric proofs are logical arguments that use trigonometric identities to demonstrate the truth of a statement. When working with proofs, like in our original exercise, it's vital to:

• Understand and apply core identities correctly
• Simplify expressions to reveal the desired result

In the exercise, using both the Angle Addition and Subtraction Formulas, we aimed to show equivalence between the initial expression and the target identity.
This proof was successful by transforming and simplifying terms step by step.
Solving such proofs strengthens understanding of how trigonometric functions interact and illustrates their relationships.
Mastery of these proofs requires practice and a thorough grasp of fundamental trigonometric properties.
They also offer a practical way of verifying the correctness of trigonometric relationships in more complex problems.