Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 64

Chapter 6, Problem 64

Verify that each equation is an identity. See Example 4.
tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

Verified step by step guidance

Recall the tangent subtraction formula: \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\). This will be useful to expand both \(\tan(x - y)\) and \(\tan(y - x)\).

Apply the formula to \(\tan(x - y)\): write it as \(\frac{\tan x - \tan y}{1 + \tan x \tan y}\).

Apply the formula to \(\tan(y - x)\): write it as \(\frac{\tan y - \tan x}{1 + \tan y \tan x}\). Note that \(1 + \tan y \tan x\) is the same as \(1 + \tan x \tan y\) due to commutativity.

Substitute these expressions back into the left side of the equation: \(\tan(x - y) - \tan(y - x) = \frac{\tan x - \tan y}{1 + \tan x \tan y} - \frac{\tan y - \tan x}{1 + \tan x \tan y}\).

Since the denominators are the same, combine the fractions and simplify the numerator: \(\frac{(\tan x - \tan y) - (\tan y - \tan x)}{1 + \tan x \tan y}\). Simplify the numerator carefully to show it equals \(2(\tan x - \tan y)\), which matches the right side of the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often using known formulas like angle sum and difference identities.

Tangent Difference Formula

The tangent difference formula states that tan(a - b) = (tan a - tan b) / (1 + tan a tan b). This formula is essential for rewriting and simplifying expressions involving tangent of differences, which is key to verifying the given identity.

Algebraic Manipulation of Fractions

Simplifying trigonometric expressions often requires careful algebraic manipulation, including combining fractions, factoring, and simplifying complex rational expressions. Mastery of these skills helps in transforming one side of the identity to match the other.

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