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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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