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 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 68


$$\sec(\pi-x)=-\sec x$$ We attempt the left side first to reach the conclusion that the equation is an identity.

Work Step by Step

$$\sec(\pi-x)=-\sec x$$ We attempt from the left side first. $$A=\sec(\pi-x)$$ It is known that $$\sec\theta=\frac{1}{\cos\theta}$$ which means $$A=\frac{1}{\cos(\pi-x)}$$ Now we expand $\cos(\pi -x)$ according to the cosine difference identity, which states $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ $$A=\frac{1}{\cos\pi\cos x+\sin\pi\sin x}$$ $$A=\frac{1}{-1\times\cos x+0\times\sin x}$$ $$A=\frac{1}{-\cos x}$$ $$A=-\frac{1}{\cos x}$$ $$A=-\sec x$$ (as $\sec\theta=\frac{1}{\cos\theta}$) The equation is true, so it is an identity.
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