Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 7 - Analytic Trigonometry - 7.4 Trigonometric Identities - 7.4 Assess Your Understanding - Page 476: 24

Answer

I know that $\cot$ is an odd function, which means $f(-\theta)=-f(\theta).$ Therefore $\cot{-\theta}=-\cot{\theta}$. Thus: $1+\cot^2{(-\theta)}=1+(-\cot{(-\theta)})^2=1+\cot^2{\theta}.$ We know that $1+\cot^2{\theta}=\csc^2{\theta}$, hence we proved the identity.

Work Step by Step

I know that $\cot$ is an odd function, which means $f(-\theta)=-f(\theta).$ Therefore $\cot{-\theta}=-\cot{\theta}$. Thus: $1+\cot^2{(-\theta)}=1+(-\cot{(-\theta)})^2=1+\cot^2{\theta}.$ We know that $1+\cot^2{\theta}=\csc^2{\theta}$, hence we proved the identity.
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