Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 7 - Chapter Test - Page 554: 7


$\dfrac{\csc \alpha+\sec \alpha }{\sin \alpha+\cos \alpha}=\cot \alpha +\tan \alpha=\dfrac{1 }{\sin \alpha+\cos \alpha}(proved)$

Work Step by Step

Prove the identity $\dfrac{\csc \alpha+\sec \alpha }{\sin \alpha+\cos \alpha}=\cot \alpha +\tan \alpha$ Consider the left side:$\dfrac{\csc \alpha+\sec \alpha }{\sin \alpha+\cos \alpha}=\dfrac{1/\sin \alpha+1/\cos \alpha }{\sin \alpha+\cos \alpha} =\dfrac{1 }{\sin \alpha+\cos \alpha}$ Now, consider the right side:$\cot \alpha +\tan \alpha=\dfrac{\cos \alpha }{\sin \alpha}+\dfrac{\sin \alpha }{\cos \alpha}=\dfrac{1 }{\sin \alpha+\cos \alpha}$ Thus, $\dfrac{\csc \alpha+\sec \alpha }{\sin \alpha+\cos \alpha}=\cot \alpha +\tan \alpha=\dfrac{1 }{\sin \alpha+\cos \alpha}$(proved)
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