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: 98


$$\sqrt{\cos^2t}=\cos t$$ Pick $t=\pi$ and replace into $\cos t$ and $\sqrt{\cos^2t}$. The results came out differently, so the equation is not an identity.

Work Step by Step

$$\sqrt{\cos^2t}=\cos t$$ To prove this is not an identity, we need to find an example that counters the equation. We can choose $t=\pi$. At $t=\pi$, $$\cos t=\cos\pi=-1$$ while at the same time, $$\sqrt{\cos^2t}=\sqrt{\cos^2\pi}=\sqrt{(-1)^2}=\sqrt1=1$$ Since $-1\ne1$, that means $\cos t\ne\sqrt{\cos^2t}$ at $t=\pi$. The equation, thus, cannot be an identity.
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