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

Double Angle Identities

Trigonometry

6. Trigonometric Identities and More Equations

Double Angle Identities

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Problem 5.48

Textbook Question

Simplify each expression. See Example 4.
cos² 2x - sin² 2x

Verified step by step guidance

Recognize that the expression \( \cos^2 2x - \sin^2 2x \) is a form of the trigonometric identity for the cosine of a double angle.

Recall the double angle identity: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).

Identify that in this problem, \( \theta = 2x \), so the expression \( \cos^2 2x - \sin^2 2x \) can be rewritten using the identity.

Apply the identity: \( \cos 2(2x) = \cos^2 2x - \sin^2 2x \).

Simplify the expression to \( \cos 4x \) using the double angle identity.

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Key Concepts

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

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship between sine and cosine functions is essential in trigonometry, allowing for the simplification of expressions involving these functions. Understanding this identity helps in transforming and simplifying trigonometric expressions effectively.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles in terms of single angles. For example, cos(2x) can be expressed as cos²(x) - sin²(x) or 2cos²(x) - 1. These formulas are crucial for simplifying expressions like cos²(2x) - sin²(2x) by rewriting them in a more manageable form.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In trigonometry, this includes factoring, combining like terms, and applying identities. Mastery of algebraic manipulation is vital for simplifying trigonometric expressions, as it allows for the application of identities and the reduction of complex terms into simpler forms.