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

Verified step by step guidance

1. Recognize that the expression is in the form of a trigonometric identity. Specifically, it resembles the identity for the cosine of a double angle: \( \cos(2\theta) = 1 - 2\sin^2(\theta) \).
2. Identify \( \theta \) in the expression. Here, \( \theta = 22.5^\circ \).
3. Apply the double angle identity: \( 1 - 2\sin^2(22.5^\circ) = \cos(2 \times 22.5^\circ) \).
4. Calculate the angle: \( 2 \times 22.5^\circ = 45^\circ \).
5. Substitute back into the expression: \( \cos(45^\circ) \).

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship allows us to express one trigonometric function in terms of another, which is essential for simplifying expressions involving sine and cosine.

Double Angle Formulas

Double angle formulas provide relationships for trigonometric functions of double angles, such as sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ). These formulas can be useful in simplifying expressions that involve angles that are multiples of a given angle.

Trigonometric Values of Special Angles

Certain angles, such as 0°, 30°, 45°, 60°, and 90°, have known sine and cosine values. For example, sin(30°) = 1/2 and cos(30°) = √3/2. Knowing these values can help in simplifying expressions involving trigonometric functions at these specific angles.