Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin x+\sin y}{\cos x-\cos y}=-\cot \frac{x-y}{2}$$

Step-by-Step Solution

Answer

After simplification, both sides of the equation \(\frac{\sin x+\sin y}{\cos x-\cos y}=-\cot \frac{x-y}{2}\) are found to be identical, thus the identity is verified.

Step 1: Expand and Simplify the Right Hand Side

Rewrite \(-\cot \frac{x-y}{2}\) to \(-\frac{\cos \frac{x-y}{2}}{\sin \frac{x-y}{2}}\)

Step 2: Use Double-Angle Formulas

Using the double-angle formulas, where \(\cos(2\theta) = 1 - 2\sin^2(\theta)\),in the denominator, and \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), in the numerator, we can rewrite the equation as \(-\frac{2\sin\frac{x}{2}\cos\frac{x}{2}-2\sin\frac{y}{2}\cos\frac{y}{2}}{1-2\sin^2\frac{x}{2}-1+2\sin^2\frac{y}{2}}\)

Step 3: Simplify the Expression Further

Simplify the expression to yield \(-\frac{\sin x - \sin y}{\cos x - \cos y}\)

Step 4: Expand and Simplify the Left Hand Side

The left-hand side of the given equation is \(\frac{\sin x+\sin y}{\cos x-\cos y}\) which is already in the simplified form.

Step 5: Compare the Right Hand Side and Left Hand Side

After simplifying, it's found that both sides are identical, thus verifying the identity algebraically.

Key Concepts

Double-Angle Formulas

When dealing with trigonometric identities, double-angle formulas can be very useful. These formulas allow us to express trigonometric functions of double angles, such as \( \sin(2\theta) \) and \( \cos(2\theta) \), in terms of single angles. For example:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1 \)

These equations are particularly useful for simplifying expressions or verifying identities, as they allow us to replace complex trigonometric terms with simpler expressions. In the context of the problem at hand, by applying these identities, we transform complex functions into simpler terms that are much easier to handle and compare. In this specific exercise, the double-angle formulas are employed to simplify the equation and help verify the trigonometric identity algebraically.

Simplifying Expressions

Simplifying expressions in trigonometry is a critical step in verifying identities or solving equations. The goal is to transform complex expressions into simpler, more manageable forms without changing their value. This process often involves using known trigonometric identities and properties, such as the Pythagorean identity or angle sum and difference identities.

In the exercise, the expression \( -\cot \frac{x-y}{2} \) is simplified to \( -\frac{\cos \frac{x-y}{2}}{\sin \frac{x-y}{2}} \). Through the steps given, the application of double-angle formulas and algebraic manipulation allows further simplification. Eventually, the expression becomes identical to another form, verifying the identity-algebraically.

Understanding how to effectively simplify expressions requires practice and familiarity with trigonometric properties, but it is an essential skill when dealing with trigonometry problems.

Graphing Utilities

Graphing utilities are tools that can graph mathematical functions and equations, aiding in visually verifying algebraic work. In trigonometry, graphing utilities can be invaluable for comparing graphs to see if two sides of an equation are indeed equivalent.

• They allow students to visualize trigonometric identities and understand the behavior of functions.
• They offer immediate feedback, showing whether transformations or simplifications lead to the same function.

In this exercise, after algebraically verifying the identity, a graphing utility can be used to plot both sides of the equation separately. If the graphs overlap completely, it supports the algebraic proof, showing that both expressions represent the same trigonometric function or relationship. Thus, graphing utilities provide a graphical confirmation of our analytical results, reinforcing understanding through visualization.