Problem 40 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\cos x - \cos y}{\sin x + \sin y} + \dfrac{\sin x - \sin y}{\cos x + \cos y} = 0 \)

Step-by-Step Solution

Answer

0

Step 1: Recall the trigonometric identities

We first need to recall the trigonometric identities: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \) and \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). We will use these identities to simplify both fractions.

Step 2: Apply the trigonometric identities

Now, apply the given identities to both fractions. For the first fraction, we rewrite as \( \dfrac{\cos x \cos y + \sin x \sin y - 2 \cos x \cos y}{\sin x \cos y + \cos x \sin y + \sin y \cos x + \sin x \sin y} \) which simplifies to \( \dfrac{\cos(y - x) - \cos x \cos y}{\sin(y + x)} \). For the second fraction we rewrite as \( \dfrac{\sin x \cos y - \cos x \sin y - 2 \sin x \sin y}{\cos x \cos y + \sin x \sin y + \cos y \cos x + \sin y \sin x} \) which simplifies to \( \dfrac{\sin(y - x) - \sin x \sin y}{\cos(y + x)} \).

Step 3: Combine the two fractions

We now add both simplified fractions. We have \( \dfrac{\cos(y - x) - \cos x \cos y}{\sin(y + x)} + \dfrac{\sin(y - x) - \sin x \sin y}{\cos(y + x)} \). Next we add the numerators and denominators so we get \( \dfrac{\cos(y - x) + \sin(y - x) - (\cos x \cos y + \sin x \sin y)}{\sin(y + x) + \cos(y + x)} \).

Step 4: Simplify the combined fraction

Simplify the fraction by using the trigonometric identity for sum and difference of two angles, \( \sin(y + x) + \cos(y + x) = 1 \). Also note that the numerator simplifies to \( \cos(y - x) + \sin(y - x) - 1 \). So we get \( \dfrac{\cos(y - x) + \sin(y - x) - 1}{1} = \cos(y - x) + \sin(y - x) - 1 \). We know that both \( \cos(y - x) \) and \( \sin(y - x) \) are between -1 and 1. Therefore, the entire expression is always 0 and the identity is indeed verified.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial in simplifying complex trigonometric expressions and solving trigonometric equations. At their core, trigonometric identities are equations that are true for all values of the variables involved.

Common trigonometric identities include reciprocal identities, quotient identities, Pythagorean identities, and even-odd identities. For example, the reciprocal identity for sine and cosecant is expressed as \( \frac{1}{\text{sin } x} = \text{csc } x \).

In the exercise provided, we specifically use the cosine of a difference identity, \( \text{cos}(a - b) = \text{cos } a \text{ cos } b + \text{sin } a \text{ sin } b \), and the sine of a difference identity, \( \text{sin}(a - b) = \text{sin } a \text{ cos } b - \text{cos } a \text{ sin } b \), to simplify the expression. These particular identities are invaluable when you encounter sum or difference of angles within the trigonometric functions.

Sum and Difference Identities

The sum and difference identities are a subset of trigonometric identities that deal with the sine, cosine, and tangent of the sum or difference of two angles. They are the foundation for simplifying expressions where angles are being added or subtracted within a trigonometric function.

For sine, we have \( \text{sin}(x \text{ + } y) = \text{sin } x \text{ cos } y + \text{cos } x \text{ sin } y \) and \( \text{sin}(x \text{ - } y) = \text{sin } x \text{ cos } y - \text{cos } x \text{ sin } y \). Similarly, for cosine, \( \text{cos}(x \text{ + } y) = \text{cos } x \text{ cos } y - \text{sin } x \text{ sin } y \) and \( \text{cos}(x \text{ - } y) = \text{cos } x \text{ cos } y + \text{sin } x \text{ sin } y \).

When proceeding with verification or simplification, as in the given exercise, these identities allow us to transform the expression in a way that showcases the inherent relationships between the trigonometric functions of the various angle combinations.

Simplifying Trigonometric Expressions

To simplify a trigonometric expression means to rewrite it in a more concise and elegant form. This usually involves using trigonometric identities to combine and reduce the expression to its simplest terms.

In simplifying an expression, it's essential to look out for opportunities to factor, to cancel out terms, and to combine like terms. Consider if the given problem can be restructured using known identities, just as the exercise we're addressing seeks to demonstrate.

Here, both sum and difference identities are applied. In the process of simplifying the provided trigonometric expression, we look for ways to rewrite each term to reveal simplifications that confirm the identity equals zero. This process not only verifies the identity at hand but also reinforces the interconnected nature of trigonometric functions and their identities.