Problem 39 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\frac{\cos (x-y)}{\cos x \cos y}=1+\tan x \tan y$$

Step-by-Step Solution

Answer

Based on the step-by-step solution above, answer the following short question: **Question:** Prove the trigonometric identity: \(\frac{\cos (x-y)}{\cos x \cos y}=1+\tan x \tan y\) **Answer:** To prove this identity, substitute the angle difference formula for cosine in the numerator, which gives: $$\frac{\cos (x-y)}{\cos x \cos y} = \frac{\cos x \cos y + \sin x \sin y}{\cos x \cos y}$$ Next, simplify the left side by separating the two terms in the numerator into two fractions: $$1 + \frac{\sin x \sin y}{\cos x \cos y}$$ Lastly, transform the second term to tangents using the definition of tangent: $$1 + \frac{\sin x \sin y}{\cos x \cos y} = 1 + \tan x \tan y$$ Thus, we have successfully proved the given trigonometric identity: $$\frac{\cos (x-y)}{\cos x \cos y}=1+\tan x \tan y$$

Step 1: Substitute Angle Difference Formula

Replace the numerator on the left side of the equation with the angle difference formula for cosine: $$\frac{\cos (x-y)}{\cos x \cos y} = \frac{\cos x \cos y + \sin x \sin y}{\cos x \cos y}$$

Step 2: Simplify the Left Side

To simplify the left side of the equation, separate the two terms in the numerator into two fractions: $$\frac{\cos x \cos y + \sin x \sin y}{\cos x \cos y} = \frac{\cos x \cos y}{\cos x \cos y} + \frac{\sin x \sin y}{\cos x \cos y}$$ Now, notice that the first term simplifies to \(1\): $$1 + \frac{\sin x \sin y}{\cos x \cos y}$$

Step 3: Transform the Second Term

To transform the second term to tangents, we first observe that we can rewrite \(\sin x \sin y\) as the product of two tangents using the definition of tangent: $$\tan x \cdot \tan y = \frac{\sin x}{\cos x} \cdot \frac{\sin y}{\cos y} = \frac{\sin x \sin y}{\cos x \cos y}$$ Now, substitute the \(\tan x \tan y\) back into the equation: $$1 + \frac{\sin x \sin y}{\cos x \cos y} = 1 + \tan x \tan y$$

Step 4: Conclusion

We have successfully proved that the given trigonometric identity holds: $$\frac{\cos (x-y)}{\cos x \cos y}=1+\tan x \tan y$$

Key Concepts

Angle Difference Formula

Trigonometry gives us several powerful tools, and the Angle Difference Formula for cosine is one of them. It allows us to express the cosine of the difference of two angles, say \( x \) and \( y \), as:

• \( \cos(x-y) = \cos x \cos y + \sin x \sin y \).

This formula is essential when dealing with trigonometric identities because it lets us break down complex expressions into more manageable pieces. In the given exercise, substituting \( \cos(x-y) \) with its formula helps in transforming the identity into fractions that are more straightforward to work with. By turning complex expressions into sums or differences, we can often find ways to simplify or prove what we set out to demonstrate.

Simplifying Trigonometric Expressions

The simplification of trigonometric expressions is a common pursuit in mathematics, providing clarity and ease in solving problems. Once we substitute the Angle Difference Formula, simplification occurs by breaking apart the numerator, which was originally \( \cos x \cos y + \sin x \sin y \), over the common denominator \( \cos x \cos y \). This process yields:

• \( \frac{\cos x \cos y}{\cos x \cos y} + \frac{\sin x \sin y}{\cos x \cos y} \).

Notice how each component now stands on its own, making it easier to recognize the first term simplifies directly to 1. The process of splitting and simplifying such expressions is vital because it converts a potentially bulky problem into smaller, more recognizable parts that we can handle with standard trigonometric properties.

Tangent Function

The tangent function, \( \tan x \), is an integral part of trigonometry, representing the ratio of sine to cosine:

• \( \tan x = \frac{\sin x}{\cos x} \).

In the context of the given identity, transforming the second fraction involves recognizing this ratio. We see \( \frac{\sin x \sin y}{\cos x \cos y} \) aligns with the product \( \tan x \tan y \). Hence, replacing the quotient with the product of tangents allows us to neatly describe the relationship between the sine and cosine components of two angles. The tangent function simplifies expressions significantly by reducing them to just one term, facilitating easier manipulation and proofs.

Proving Identities

Proving trigonometric identities involves showing that one side of an equation can transform into another. Initially, it may seem difficult, but methods such as substitution, simplification, and recognition of fundamental trigonometric functions turn this into an achievable task. In this exercise, we begin by applying the Angle Difference Formula to decompose cosine differences, then simplify using algebraic separation, and identify tangent forms within the expressions.

• By confirming both sides of the identity ultimately match, we've completed the proof.

Proving trigonometric identities not only solidifies understanding of trigonometric functions but also enhances algebraic problem-solving skills. It requires a blend of strategic thinking and technical finesse to ascertain correctness and reach that "aha" moment when both sides align perfectly.