Problem 23 - Contemporary Precalculus

Question

Simplify the given expression. $$\cos (x+y)-\cos (x-y)$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\cos (x+y)-\cos (x-y)$$ using the sum to product identities. Answer: The simplified expression is $$-2\sin x \sin y$$.

Step 1: Recognize the formula

Identify the given expression with the sum and difference of angles $$\cos(A+B) - \cos(A-B)$$ where A = x and B = y.

Step 2: Apply the sum to product identity

Use the second sum to product identity, $$\cos(A+B) - \cos(A-B) = -2\sin A \sin B$$, and substitute A = x and B = y to get the following expression: $$\cos(x+y) - \cos(x-y) = -2\sin x \sin y$$ Therefore, the simplified expression is: $$-2\sin x \sin y$$

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions of angles to one another. These identities are fundamental tools in trigonometry and are used to simplify complex trigonometric expressions, making them easier to work with.

• They exist because of the inherent relationships between the sides and angles of right triangles.
• Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).
• More advanced identities involve multiple angles or products, such as sum-to-product identities.

Trigonometric identities are crucial for solving equations, proving other mathematical theorems, and modeling periodic phenomena in the real world. The identity used in this exercise is part of the sum-to-product category, which converts sums or differences of trigonometric functions into products.

Understanding Angle Sum and Difference

The angle sum and difference identities form another key aspect of trigonometry. These formulas allow us to express trigonometric functions of sums or differences of angles in terms of the functions of individual angles. For the cosine function, the identities are:

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

Understanding these allows you to analyze and simplify expressions like \(\cos(x+y) - \cos(x-y)\). In this exercise, knowing these identities makes it possible to apply the sum-to-product transformation efficiently. This approach highlights how various trigonometric formulas interplay, each technique building on another, ultimately leading to a simplified result.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a fundamental skill in mathematics. By using identities, we can transform complex expressions into simpler ones. This not only makes calculations more manageable but also reveals deeper relationships between trigonometric functions.

• Consider the expression \(\cos(x+y) - \cos(x-y)\). At first glance, it seems complex, but it can be simplified using trigonometric identities.
• Here, the sum-to-product identity \(\cos(A+B) - \cos(A-B) = -2\sin A \sin B\) allows the transformation of the expression to \(-2\sin x \sin y\).
• This transformation involves recognizing patterns and applying the appropriate identity to reach a simpler form.

Successful simplification requires practice and familiarity with various identities. With persistence, you'll find these tasks become second nature, enhancing your ability to tackle more complex problems in trigonometry and beyond.