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 5.5 Double-Angle Identities - 5.5 Exercises - Page 237: 28


$$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$ 2 sides are equal as shown in the work step by step, and the equation is an identity.

Work Step by Step

$$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$ This time we do a little bit differently. We take from the left side. $$X=\cot4\theta$$ As in Reciprocal Identities: $\cot\theta=\frac{1}{\tan\theta}$, it means $\cot4\theta=\frac{1}{\tan4\theta}$ $$X=\frac{1}{\tan4\theta}$$ $$X=\frac{1}{\tan(2\times2\theta)}$$ For $\tan(2\times2\theta)$, we apply Double-Angle Identity for $\tan2A$, which states $$\tan2A=\frac{2\tan A}{1-\tan^2A}$$ With $A=2\theta$, we have $$\tan(2\times2\theta)=\frac{2\tan2\theta}{1-\tan^22\theta}$$ Thus, $$X=\frac{1}{\frac{2\tan2\theta}{1-\tan^22\theta}}$$ $$X=\frac{1-\tan^22\theta}{2\tan2\theta}$$ Therefore, $$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$ 2 sides are thus equal, and the equation is an identity.
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