Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.6 Half-Angle Identities - 5.6 Exercises - Page 243: 41



Work Step by Step

$$\pm\sqrt{\frac{1-\cos8\theta}{1+\cos8\theta}}$$ Here we notice at the start of the expression, there is the sign $\pm$, which means we do not have to decide whether to take positive or negative square root. From the half-angle identity for tangents: $$\pm\sqrt{\frac{1-\cos A}{1+\cos A}}=\tan\frac{A}{2}$$ We can apply the identity to the given expression with $A=8\theta$. $$\pm\sqrt{\frac{1-\cos8\theta}{1+\cos8\theta}}=\tan\frac{8\theta}{2}$$ $$\pm\sqrt{\frac{1-\cos8\theta}{1+\cos8\theta}}=\tan4\theta$$
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