Simplify each expression. See Example 4.
1 - 2 sin² 22 ½°

Verified step by step guidance

1. Recognize that the expression involves a trigonometric identity related to sine squared. Recall the double-angle identity for cosine: \(\cos(2\theta) = 1 - 2\sin^{2}(\theta)\).
2. Identify the angle in the expression: here, \(\theta = 22 \frac{1}{2}^\circ\) (which is \(22.5^\circ\)).
3. Rewrite the expression \(1 - 2\sin^{2}(22.5^\circ)\) using the double-angle identity: it equals \(\cos(2 \times 22.5^\circ)\).
4. Calculate the angle inside the cosine function: \(2 \times 22.5^\circ = 45^\circ\).
5. Therefore, the expression simplifies to \(\cos(45^\circ)\), which is a well-known exact value.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This fundamental relationship allows us to express sine squared terms in terms of cosine squared, and vice versa, which is useful for simplifying trigonometric expressions.

Double-Angle Formula for Cosine

The double-angle formula for cosine is cos(2θ) = 1 - 2sin²θ. This formula directly relates sin²θ to cos(2θ), enabling simplification of expressions involving sin²θ by rewriting them in terms of cosine of a double angle.

Angle Conversion and Notation

Understanding angle notation, such as 22 ½° (which is 22.5° or 22.5 degrees), is essential for applying formulas correctly. Recognizing this angle helps in substituting values or using known exact trigonometric values for simplification.