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: 31


- Apply Addition Formula for cosine to the left side. - Simplify. Then, we have the identity: $$\cos\Big(x-\frac{\pi}{2}\Big)=\sin x$$

Work Step by Step

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