Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 4 - Section 1.5 - More on Identities - 1.5 Problem Set - Page 47: 82


Showed that given statement, $\sec\theta -\cos\theta$ = $\frac{\sin^{2}\theta}{\cos\theta}$, is an identity as left side transforms into right side.

Work Step by Step

Given statement is- $\sec\theta -\cos\theta$ = $\frac{\sin^{2}\theta}{\cos\theta}$ Left Side = $\sec\theta -\cos\theta$ = $\frac{1}{\cos\theta} -\cos\theta$ ( Using reciprocal identity for $\sec\theta$) = $\frac{1}{\cos\theta} - \cos\theta\times\frac{\cos\theta}{\cos\theta}$ = $\frac{1}{\cos\theta} - \frac{\cos^{2}\theta}{\cos\theta}$ = $\frac{1-\cos^{2}\theta}{\cos\theta}$ = $\frac{\sin^{2}\theta}{\cos\theta}$ [ From first Pythagorean identity, $ (1 - \cos^{2}\theta)$ can be written as $\sin^{2}\theta$] = Right Side i.e. Left Side transforms into Right Side i.e. Given statement, $\sec\theta -\cos\theta$ = $\frac{\sin^{2}\theta}{\cos\theta}$, is an identity.
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