Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - Summary Exercises Verifying Trigonometric Identities - Exercises: 19

Answer

$\tan 4\theta =\dfrac {2\tan 2\theta }{2-sec^22\theta }$

Work Step by Step

$\tan 4\theta =\tan \left( 2\theta +2\theta \right) =\dfrac {\tan 2\theta +\tan 2\theta }{1-\left( \tan 2\theta \right) \times \left( \tan 2\theta \right) }=\dfrac {2\tan 2\theta }{1-\tan ^{2}2\theta }=\dfrac {2\tan \theta }{1-\dfrac {\sin ^{2}2\theta }{\cos ^{2}2\theta }}=\dfrac {\dfrac {2\tan 2\theta }{\cos ^{2}2\theta -\left( 1-\cos ^{2}2\theta \right) }}{\cos ^{2}2\theta }=\dfrac {2\tan 2\theta }{2-\dfrac {1}{\cos ^{2}2\theta }}=\dfrac {2\tan 2\theta }{2-sec^22\theta }$
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