Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 203: 71

Answer

$$(1-\cos^2\alpha)(1+\cos^2\alpha)=2\sin^2\alpha-\sin^4\alpha$$ By dealing with the left side. we prove that both sides are equal and this is thus an identity.

Work Step by Step

$$(1-\cos^2\alpha)(1+\cos^2\alpha)=2\sin^2\alpha-\sin^4\alpha$$ We would try with the left side first. $$A=(1-\cos^2\alpha)(1+\cos^2\alpha)$$ We notice that $\sin^2\alpha=1-\cos^2\alpha$. That means, $$A=\sin^2\alpha(1+\cos^2\alpha)$$ $$A=\sin^2\alpha+\sin^2\alpha\cos^2\alpha$$ Also, as we witness that the right side only includes $\sin\alpha$, it is better to change $\cos^2\alpha$ into $1-\sin^2\alpha$. $$A=\sin^2\alpha+\sin^2\alpha(1-\sin^2\alpha)$$ $$A=\sin^2\alpha+\sin^2\alpha-\sin^4\alpha$$ $$A=2\sin^2\alpha-\sin^4\alpha$$ Thus, the left side is equal to the right side. The expression is therefore an identity.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.