University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.3 - Trigonometric Functions - Exercises - Page 28: 35


- Apply the Addition Formula for cosine to the left side. - Then simplify. 2 sides would be equal, thus we have the identity: $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$

Work Step by Step

*Addition Formula for cosine: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ The identity needed to prove here: $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ *Consider the left side and apply Addition Formula here: $$\cos(A-B)=\cos[A+(-B)]$$ $$\cos(A-B)=\cos A\cos(-B)-\sin A\sin(-B)$$ We have $\cos(-B)=\cos B$ and $\sin(-B)=-\sin B$ (because cosine is an even function, and sine is an odd one) Therefore, $$\cos(A-B)=\cos A\cos B-\sin A(-\sin B)$$ $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ The identity has been proved.
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