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 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 228: 62


$$\sin(x+y)+\sin(x-y)=2\sin x\cos y$$ The equation has been verified to be an identity.

Work Step by Step

$$\sin(x+y)+\sin(x-y)=2\sin x\cos y$$ Let's examine the left side. $$X=\sin(x+y)+\sin(x-y)$$ According to the sine sum and difference identities: $$\sin(A+B)=\sin A\cos B+\cos A\sin B$$ $$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ Therefore, $$X=(\sin x\cos y+\cos y\sin x)+(\sin x\cos y-\cos y\sin x)$$ $$X=(\sin x\cos y+\sin x\cos y)+(\cos y\sin x-\cos y\sin x)$$ $$X=2\sin x\cos y$$ Thus 2 sides are equal. This equation has been verified to be an identity.
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