Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 210: 99


$$\csc t=\sqrt{1+\cot^2 t}$$ After taking $t=-\frac{\pi}{2}$, we find that $\csc t\ne\sqrt{1+\cot^2t}$ at $t=-\frac{\pi}{2}$. That means the equation is not an identity.

Work Step by Step

$$\csc t=\sqrt{1+\cot^2 t}$$ We would find an example that disproves the equation, thus showing that it is not an identity. Pick $t=-\frac{\pi}{2}$. That means, $$\csc t=\csc\Big(-\frac{\pi}{2}\Big)=-1$$ and $$\sqrt{1+\cot^2t}=\sqrt{1+\cot^2\Big(-\frac{\pi}{2}\Big)}=\sqrt{1+0^2}=\sqrt1=1$$ $-1\ne1$, which means $\csc t\ne\sqrt{1+\cot^2t}$ at $t=-\frac{\pi}{2}$ The equation is not an identity.
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