Problem 82 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\tan x+\tan y}{\cot x+\cot y}=\frac{\tan x \tan y-1}{1-\cot x \cot y}$$

Step-by-Step Solution

Answer

Question: Prove the identity $\frac{\tan x\tan y}{\tan x + \tan y} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$. Answer: We have proved the given identity by converting cotangents to tangents, simplifying the numerator and denominator, and canceling out factors where possible. The detailed step-by-step solution is shown above.

Step 1: Convert cotangents to tangents

Recall that \(\cot x = \frac{1}{\tan x}\). We can use this relationship to rewrite the expression containing cotangents in terms of tangents only. So, let's replace all instances of \(\cot x\) and \(\cot y\) with \(\frac{1}{\tan x}\) and \(\frac{1}{\tan y}\), respectively: $$\frac{\tan x+\tan y}{\frac{1}{\tan x}+\frac{1}{\tan y}}=\frac{\tan x \tan y - 1}{1 - \frac{1}{\tan x}\cdot\frac{1}{\tan y}}$$

Step 2: Simplify the denominator

In the left-hand side of the equation, the denominator has the same structure as a common denominator for fractions. Therefore, we can simplify it by finding a common denominator. The common denominator between \(\tan x\) and \(\tan y\) is their product, \(\tan x\tan y\). So, let's rewrite the denominator in terms of this common denominator: $$\frac{\tan x + \tan y}{\frac{\tan x\tan y}{\tan x}+\frac{\tan x\tan y}{\tan y}} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$

Step 3: Simplify the numerator

Now, we can also simplify the numerator of the left-hand side of the equation: $$\frac{\tan x+\tan y}{\frac{\tan x\tan y}{\tan x}+\frac{\tan x\tan y}{\tan y}} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$ Expanding the numerator, we get: $$\frac{\tan x\tan y +\tan y\tan x}{\frac{\tan x\tan y}{\tan x}+\frac{\tan x\tan y}{\tan y}} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$ Combining the two tangent terms gives: $$\frac{2\tan x\tan y}{\frac{\tan x\tan y}{\tan x}+\frac{\tan x\tan y}{\tan y}} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$

Step 4: Simplify the denominator further

Let's simplify the right-hand side of the equation by multiplying the denominator by \(\tan x\tan y\): $$\frac{2\tan x\tan y}{\frac{\tan x\tan y}{\tan x}+\frac{\tan x\tan y}{\tan y}} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$ Expanding the denominator, we get: $$\frac{2\tan x\tan y}{\tan y+\tan x} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$ And finally, we can cancel the factors of 2 in the numerator and denominator of the left-hand side to obtain the desired identity: $$\frac{\tan x\tan y}{\tan x + \tan y} = \frac{\tan x\tan y - 1}{1 - \frac{1}{\tan x\tan y}}$$ Thus, we have proved the given identity.

Key Concepts

Understanding Cotangent

The cotangent function, represented as \( \cot x \), is the reciprocal of the tangent function. This means that \( \cot x = \frac{1}{\tan x} \). This relationship is essential when working with trigonometric identities since it allows us to convert equations involving cotangent into expressions involving tangent. By converting an equation involving \( \cot x \) to \( \frac{1}{\tan x} \), it simplifies the process of proving trigonometric identities by working with familiar tangent terms instead.

Using these relationships can make solving identities easier as it reduces the complexity of the equation.

• Recognize \( \cot x = \frac{1}{\tan x} \) and \( \cot y = \frac{1}{\tan y} \).
• Transform cotangent expressions into tangent for uniformity in equations.

Exploring Tangent

The tangent function, denoted as \( \tan x \), is a fundamental part of trigonometry. It's defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Another way to express tangent is through sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \).

Tangent plays a significant role in simplifying trigonometric identities. By understanding how \( \tan x \) interacts with other trigonometric functions, you can break down complex identities into simpler parts. This often involves rewriting other trigonometric functions in terms of tangent.

• Recall \( \tan x = \frac{\sin x}{\cos x} \).
• Convert identities involving tangent using reciprocal relationships.
• Simplify expressions by working with tangent, making complex identities easier to handle.

Proving Trigonometric Identities

Proving trigonometric identities involves establishing that two sides of an equation are equivalent. This task requires a thorough understanding of trigonometric properties and relationships. The goal is to manipulate one or both sides using algebraic and trigonometric identities until they appear identical.

Here are some steps to guide you:

• Substitute equivalent trigonometric functions (e.g., \( \cot x = \frac{1}{\tan x} \)).
• Identify common denominators to combine fractions.
• Apply known identities like Pythagorean identities, reciprocal identities, and co-function identities to simplify expressions.
• Independently manipulate both sides until they match perfectly.

These steps require practice and a clear understanding of trigonometric identities to efficiently prove them.

Trigonometric Simplification Techniques

Trigonometric simplification is a vital skill that involves breaking down complex expressions into simpler terms. By simplifying trigonometric expressions, we can prove identities more easily or solve equations more effectively.

Here are some key techniques:

• Convert all terms to a single trigonometric function like tangent to unify the expression.
• Use reciprocal, quotient, and Pythagorean identities to analyze and simplify terms.
• Look for opportunities to factor expressions or combine like terms.
• Simplify complex fractions by common denominators or by removing negative exponents.

By mastering these simplification techniques, you can more easily tackle problems involving trigonometric identities and equations. Simplification often leads to a clearer path towards solving or proving a given problem efficiently.