Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2

Verified step by step guidance

1. Recognize that the expression \( \cot^4 x + 3 \cot^2 x + 2 \) can be treated as a quadratic in terms of \( \cot^2 x \).
2. Let \( u = \cot^2 x \). Then the expression becomes \( u^2 + 3u + 2 \).
3. Factor the quadratic expression \( u^2 + 3u + 2 \) by finding two numbers that multiply to 2 and add to 3.
4. The numbers 1 and 2 satisfy these conditions, so the expression factors as \( (u + 1)(u + 2) \).
5. Substitute back \( u = \cot^2 x \) to get the factored form \( (\cot^2 x + 1)(\cot^2 x + 2) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Understanding these identities, such as the Pythagorean identities, can help simplify and manipulate trigonometric expressions. In this case, recognizing that cotangent can be expressed in terms of sine and cosine may aid in factoring the expression.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. In the given expression, recognizing it as a quadratic in terms of cot² x allows us to apply factoring techniques similar to those used in algebra.

Quadratic Form

A quadratic form is an expression that can be written in the standard form ax² + bx + c. In the context of the given expression, cot⁴ x + 3 cot² x + 2 can be treated as a quadratic in cot² x. This perspective enables the use of factoring methods to find the roots or simplify the expression effectively.