Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 7 - Analytic Trigonometry - 7.4 Trigonometric Identities - 7.4 Assess Your Understanding - Page 476: 35

Answer

Work on the left side of the identity using $\sin^2{\theta}+\cos^2\theta=1$. Refer to the step-by-step part below for the complete proof.

Work Step by Step

We have to show that: $(\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^2=2$ By evaluating the left side we get: $=(\sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta)+(\sin^2\theta-2\sin\theta\cos\theta+\cos^2\theta)\\ =2\sin^2\theta + 2\cos^2\theta\\ =2(\sin^2\theta+\cos^2\theta)$ Using the fact that $\sin^2\theta+\cos^2\theta=1$, the above expression simplifies to $=2(1)\\ =2$ Since LHS = RHS, the identity's proof is complete.
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