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

Verified step by step guidance

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

Key Concepts

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.