Determine whether each statement is true or false. If false, tell why. See Example 4. cos 60° = 2 cos² 30° - 1

Verified step by step guidance

1. Recall the double-angle identity for cosine: \(\cos(2\theta) = 2\cos^{2}(\theta) - 1\).
2. Identify the angle in the problem: here, \(60^\circ\) is given, and the right side uses \(\cos^{2}(30^\circ)\).
3. Check if \(60^\circ\) can be expressed as \(2 \times 30^\circ\), which it can, so the identity applies with \(\theta = 30^\circ\).
4. Substitute \(\theta = 30^\circ\) into the identity: \(\cos(60^\circ) = 2\cos^{2}(30^\circ) - 1\).
5. Since this matches the given statement exactly, conclude that the statement is true based on the double-angle formula.

Key Concepts

Cosine Double-Angle Identity

The cosine double-angle identity states that cos(2θ) = 2cos²(θ) - 1. This formula allows expressing the cosine of twice an angle in terms of the cosine of the original angle, which is essential for verifying or simplifying trigonometric expressions involving double angles.

Evaluating Trigonometric Functions at Specific Angles

Knowing exact values of trigonometric functions at common angles like 30°, 60°, and 90° is crucial. For example, cos 60° = 1/2 and cos 30° = √3/2. These values help in directly substituting and verifying the truth of trigonometric statements.

Verification of Trigonometric Equations

To determine if a trigonometric statement is true or false, substitute known values or use identities to simplify both sides. Comparing the results confirms the statement's validity or reveals why it is false, which is key in problem-solving.