Factor each trigonometric expression.
sec² θ - 1

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Recognize that the expression \( \sec^2 \theta - 1 \) is a difference of squares.

Recall the Pythagorean identity: \( \sec^2 \theta = 1 + \tan^2 \theta \).

Substitute \( \sec^2 \theta \) with \( 1 + \tan^2 \theta \) in the expression: \( (1 + \tan^2 \theta) - 1 \).

Simplify the expression: \( \tan^2 \theta \).

Factor \( \tan^2 \theta \) as \( (\tan \theta)(\tan \theta) \).

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One of the fundamental identities is the Pythagorean identity, which states that sec² θ = 1 + tan² θ. Understanding these identities is crucial for simplifying and factoring trigonometric expressions.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Common techniques include recognizing patterns such as the difference of squares, which applies to expressions like sec² θ - 1. Mastery of these techniques is essential for solving trigonometric equations efficiently.

Difference of Squares

The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is particularly useful in factoring expressions where one term is the square of a variable or function. In the case of sec² θ - 1, it can be factored as (sec θ - 1)(sec θ + 1), illustrating the application of this identity in trigonometry.

Factor each trigonometric expression.sec² θ - 1