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: 81

Answer

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

Work Step by Step

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