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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 5.26
Capitolo 6, Problema 5.26

Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²

Guida verificata passo dopo passo
1
Recognize the expression as a difference of squares: \((a^2 - b^2) = (a - b)(a + b)\). Here, \(a = \tan x + \cot x\) and \(b = \tan x - \cot x\).
Apply the difference of squares formula: \((\tan x + \cot x)^2 - (\tan x - \cot x)^2 = [(\tan x + \cot x) - (\tan x - \cot x)][(\tan x + \cot x) + (\tan x - \cot x)]\).
Simplify the first factor: \((\tan x + \cot x) - (\tan x - \cot x) = \tan x + \cot x - \tan x + \cot x = 2\cot x\).
Simplify the second factor: \((\tan x + \cot x) + (\tan x - \cot x) = \tan x + \cot x + \tan x - \cot x = 2\tan x\).
Combine the simplified factors: The expression becomes \(2\cot x \cdot 2\tan x = 4\cot x \tan x\).

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. Key identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is essential for simplifying and manipulating trigonometric expressions effectively.
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Fundamental Trigonometric Identities

Difference of Squares

The difference of squares is a fundamental algebraic identity that states a² - b² = (a - b)(a + b). This concept is crucial when factoring expressions that can be represented in this form, allowing for simplification and easier manipulation of the expression. Recognizing this pattern in trigonometric expressions is key to solving the given problem.
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Sum and Difference of Tangent

Factoring Techniques

Factoring techniques involve rewriting an expression as a product of its factors, which can simplify complex expressions and make solving equations easier. Common techniques include grouping, using special products like the difference of squares, and recognizing common factors. Mastery of these techniques is vital for effectively handling trigonometric expressions and equations.
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6:08
Factoring
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