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 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 70


$$1+\cos 2x-\cos^2x=\cos^2x$$ This equation is an identity, which can be proved by transforming the left side using the identity from Exercise 69.

Work Step by Step

$$1+\cos 2x-\cos^2x=\cos^2x$$ We examine from the left side: $$A=1+\cos 2x-\cos^2x$$ The question hints at using the result from Question 69, which states that $$\cos 2x=\cos^2x-\sin^2x$$ That means we can replace $\cos2x$ in $A$ with $\cos^2x-\sin^2x$. $$A=1+\cos^2x-\sin^2x-\cos^2x$$ $$A=1-\sin^2 x$$ Now recall that $\cos^2\theta=1-\sin^2\theta$. So, $$A=\cos^2x$$ The equation has been verified to be an identity.
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