Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)

Verified step by step guidance

1. Recall the trigonometric identity for the difference of squares of cosine and sine: \(\cos^2 x - \sin^2 x = \cos 2x\).
2. Apply this identity to the given expression by substituting \(x = \frac{\pi}{6}\), so the expression becomes \(\cos 2 \times \frac{\pi}{6} = \cos \frac{\pi}{3}\).
3. Evaluate \(\cos \frac{\pi}{3}\) by recalling the unit circle values or special angles, where \(\cos \frac{\pi}{3} = \frac{1}{2}\).
4. Match the original expression \(\cos^2 \left(\frac{\pi}{6}\right) - \sin^2 \left(\frac{\pi}{6}\right)\) with the value \(\frac{1}{2}\) from Column II.
5. Verify the result by optionally calculating \(\cos^2 \left(\frac{\pi}{6}\right)\) and \(\sin^2 \left(\frac{\pi}{6}\right)\) separately and subtracting to confirm the value.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental relationship helps simplify expressions involving sine and cosine squares by relating them to each other.

Double-Angle Formula for Cosine

The double-angle formula for cosine states that cos(2θ) = cos²θ - sin²θ. This formula allows rewriting expressions like cos²θ - sin²θ in terms of a single cosine function with double the angle, simplifying evaluation.

Exact Values of Trigonometric Functions at Special Angles

Certain angles, such as π/6 (30°), have known exact sine and cosine values: sin(π/6) = 1/2 and cos(π/6) = √3/2. Using these values enables precise calculation of trigonometric expressions without a calculator.