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: 63


The statement $$\cos70^\circ\cos20^\circ-\sin70^\circ\sin20^\circ=0$$ is true.

Work Step by Step

$$\cos70^\circ\cos20^\circ-\sin70^\circ\sin20^\circ=0$$ We examine the left side: $$X=\cos70^\circ\cos20^\circ-\sin70^\circ\sin20^\circ$$ Remember the cosine sum identity: $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ So $X$ is in fact the right side of the above identity with $A=70^\circ$ and $B=20^\circ$. Therefore, we can simplify $X$: $$X=\cos(70^\circ+20^\circ)$$ $$X=\cos90^\circ$$ $$X=0$$ That means the statement $$\cos70^\circ\cos20^\circ-\sin70^\circ\sin20^\circ=0$$ is true.
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