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: 88

Answer

Showed that given statement, $ (\cos\theta + 1) ( \cos\theta- 1)$ = - $\sin^{2}\theta$, is an identity as left side transforms into right side.

Work Step by Step

Given statement is- $ (\cos\theta + 1) ( \cos\theta- 1)$ = - $\sin^{2}\theta$ Left Side = $ (\cos\theta + 1) ( \cos\theta- 1)$ = $(\cos\theta)^{2}$ - $(1)^{2}$ [ We know that, $(a)^{2}$ - $(b)^{2}$ = $ ( a + b) ( a - b) $] = $\cos^{2}\theta - 1$ = - ($1 - \cos^{2}\theta$) = - $\sin^{2}\theta$ [ 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, $ ( \cos\theta + 1) ( \cos\theta- 1)$ = - $\sin^{2}\theta$, is an identity.
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