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: 34


- Apply the Addition Formula for sine to the left side. - Simplify. - The left side would be equal with the right one; thus, proving the identity: $$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$

Work Step by Step

*Addition Formula for sine: $$\sin(A+B)=\sin A\cos B+\cos A\sin B$$ $$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$ *Consider the left side and apply Addition Formula here: $$\sin\Big(x-\frac{\pi}{2}\Big)=\sin\Big[x+\Big(-\frac{\pi}{2}\Big)\Big]=\sin x\cos\Big(-\frac{\pi}{2}\Big)+\cos x\sin\Big(-\frac{\pi}{2}\Big)$$ $$\sin\Big(x-\frac{\pi}{2}\Big)=\sin x\times0+\cos x\times(-1)$$ $$\sin\Big(x-\frac{\pi}{2}\Big)=-\cos x$$ The identity has been proved.
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