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

Verified step by step guidance

1. Recognize that the expression \( \cos^2 2x - \sin^2 2x \) is a form of the trigonometric identity for the cosine of a double angle.
2. Recall the double angle identity: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
3. Identify that in this problem, \( \theta = 2x \), so the expression \( \cos^2 2x - \sin^2 2x \) can be rewritten using the identity.
4. Apply the identity: \( \cos 2(2x) = \cos^2 2x - \sin^2 2x \).
5. Simplify the expression to \( \cos 4x \) using the double angle identity.

Key Concepts

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.