Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Problem 5.24

Textbook Question

Factor each trigonometric expression.
sec² θ - 1

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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