Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - Chapter 7 Test Prep - Review Exercises - Page 737: 53


$2\cos A-\sec A=\cos A-\frac{\tan A}{\csc A}$

Work Step by Step

Simplify the left side: $2\cos A-\sec A$ $=2\cos A-\frac{1}{\cos A}$ $=\frac{2\cos^2 A}{\cos A}-\frac{1}{\cos A}$ $=\frac{2\cos^2 A-1}{\cos A}$ $=\frac{\cos 2A}{\cos A}$ Simplify the right side: $\cos A-\frac{\tan A}{\csc A}$ $=\cos A-\frac{\frac{\sin A}{\cos A}}{\frac{1}{\sin A}}$ $=\cos A-\frac{\sin^2 A}{\cos A}$ $=\frac{\cos^2 A}{\cos A}-\frac{\sin^2 A}{\cos A}$ $=\frac{\cos^2 A-\sin^2 A}{\cos A}$ $=\frac{\cos 2A}{\cos A}$ Since both the left side and the right side are equal to $\frac{\cos 2A}{\cos A}$, they are equal to each other, and the identity is proven.
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