Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 209: 81


$$(1-\cos^2\alpha)(1+\cos^2\alpha)=2\sin^2\alpha-\sin^4\alpha$$ 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.
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