Precalculus (6th Edition)

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

Chapter 7 - Trigonometric Identities and Equations - 7.3 Sum and Difference Identities - 7.3 Exercises - Page 680: 97


$\frac{\cos(\alpha-\beta)}{\cos\alpha\sin \beta}=\tan\alpha+\cot\beta$

Work Step by Step

Start with the left side: $\frac{\cos(\alpha-\beta)}{\cos\alpha\sin \beta}$ Use the subtraction formula for cosine: $=\frac{\cos\alpha\cos\beta+\sin\alpha\sin\beta}{\cos \alpha\sin\beta}$ Break it into two fractions and simplify: $=\frac{\cos\alpha\cos\beta}{\cos \alpha\sin\beta}+\frac{\sin\alpha\sin\beta}{\cos \alpha\sin\beta}$ $=\frac{\cos\beta}{\sin\beta}+\frac{\sin\alpha}{\cos \alpha}$ $=\cot\beta+\tan \alpha$ $=\tan\alpha+\cot\beta$
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