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: 64


The statement $$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$ is true.

Work Step by Step

$$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$ We examine the left side: $$X=\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ$$ Remember the cosine difference 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 $A=85^\circ$ and $B=40^\circ$. Therefore, we can simplify $X$: $$X=\cos(85^\circ-40^\circ)$$ $$X=\cos45^\circ$$ $$X=\frac{\sqrt2}{2}$$ That means the statement $$\cos85^\circ\cos40^\circ+\sin85^\circ\sin40^\circ=\frac{\sqrt2}{2}$$ is true.
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