In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.cos² 15° - sin² 15°

Verified step by step guidance

1. Recognize that the expression \( \cos^2 15^\circ - \sin^2 15^\circ \) can be rewritten using the double-angle identity for cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \).
2. Identify that \( \theta = 15^\circ \) in this context, so the expression becomes \( \cos(2 \times 15^\circ) \).
3. Calculate \( 2 \times 15^\circ = 30^\circ \).
4. Use the known value of \( \cos 30^\circ \), which is \( \frac{\sqrt{3}}{2} \).
5. Conclude that \( \cos^2 15^\circ - \sin^2 15^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2} \).

Key Concepts

Double-Angle Formulas

Double-angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the cosine double-angle formula states that cos(2θ) = cos²(θ) - sin²(θ). This formula is essential for simplifying expressions involving angles that are multiples of a given angle, such as 30°, 45°, or 15°.

Half-Angle Formulas

Half-angle formulas allow us to express trigonometric functions of half angles in terms of the functions of the original angle. For instance, sin(θ/2) and cos(θ/2) can be derived from the sine and cosine of θ. These formulas are particularly useful when dealing with angles that are not standard, such as 15°, as they help in calculating exact values.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship between sine and cosine is crucial for deriving other trigonometric identities and simplifying expressions. In the context of the given expression, it can be used to relate sin²(15°) to cos²(15°) and vice versa, aiding in the calculation of the exact value.