Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.5 Double-Angle Identities - 5.5 Exercises - Page 230: 4

Answer

$$\cos^2\frac{\pi}{6}-\sin^2\frac{\pi}{6}=\frac{1}{2}$$ 4 would be matched with A.

Work Step by Step

$$X=\cos^2\frac{\pi}{6}-\sin^2\frac{\pi}{6}$$ From the double-angle identities: $$\cos^2 A-\sin^2 A=\cos2A$$ Thus $X$ can be seen here as $\cos^2 A-\sin^2 A$ with $A=\frac{\pi}{6}$. Therefore, $$X=\cos\Big(2\times\frac{\pi}{6}\Big)$$ $$X=\cos\frac{\pi}{3}$$ $$X=\frac{1}{2}$$ So, $$\cos^2\frac{\pi}{6}-\sin^2\frac{\pi}{6}=\frac{1}{2}$$ 4 would be matched with A.
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