Simplify each expression. See Example 4.
cos² π/8 - 1/2

Verified step by step guidance

1. Recognize that the expression involves a trigonometric identity. Specifically, use the double angle identity for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \).
2. Set \( \theta = \frac{\pi}{8} \). Then, \( 2\theta = \frac{\pi}{4} \).
3. Substitute \( \theta \) into the double angle identity: \( \cos\left(\frac{\pi}{4}\right) = 2\cos^2\left(\frac{\pi}{8}\right) - 1 \).
4. Solve for \( \cos^2\left(\frac{\pi}{8}\right) \) by rearranging the equation: \( \cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \cos\left(\frac{\pi}{4}\right)}{2} \).
5. Substitute \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \) into the equation and simplify to find \( \cos^2\left(\frac{\pi}{8}\right) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. Understanding these identities is essential for simplifying trigonometric expressions, as they provide relationships between different trigonometric functions.

Cosine Function

The cosine function, denoted as cos(θ), is a fundamental trigonometric function that relates the angle θ to the ratio of the adjacent side to the hypotenuse in a right triangle. It is also defined on the unit circle, where cos(θ) represents the x-coordinate of a point on the circle. Familiarity with the properties and values of the cosine function at specific angles is crucial for simplifying expressions involving cos²(θ).

Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For example, the cosine half-angle formula states that cos(θ/2) = ±√((1 + cos(θ))/2). These formulas are particularly useful for simplifying expressions involving angles like π/8, as they allow for the calculation of trigonometric values at half-angles.