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

Verified step by step guidance

1. Recognize that the expression \( \cos^2 2x - \sin^2 2x \) matches the form of the cosine double-angle identity, which states \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
2. Identify that in this problem, the angle \( \theta \) corresponds to \( 2x \), so the expression can be rewritten using the identity as \( \cos(2 \times 2x) \).
3. Simplify the angle inside the cosine function: \( 2 \times 2x = 4x \), so the expression becomes \( \cos 4x \).
4. Thus, the original expression \( \cos^2 2x - \sin^2 2x \) simplifies to \( \cos 4x \).
5. This shows how using trigonometric identities can transform expressions into simpler or more useful forms.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental relationship helps in simplifying trigonometric expressions by converting between sine and cosine terms.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2x in terms of functions of x. For cosine, cos 2x = cos²x - sin²x, which is directly related to the given expression and aids in simplification.

Difference of Squares in Trigonometry

The expression cos² 2x - sin² 2x resembles a difference of squares, which can be factored or recognized as a double-angle identity. Understanding this pattern allows for rewriting the expression in a simpler form.