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 - Summary Exercises on Verifying Trigonometric Identities - Page 245: 11


$$\frac{\sin(x+y)}{\cos(x-y)}=\frac{\cot x+\cot y}{1+\cot x\cot y}$$ The equation is an identity, as proved below.

Work Step by Step

$$\frac{\sin(x+y)}{\cos(x-y)}=\frac{\cot x+\cot y}{1+\cot x\cot y}$$ We start from the right side in this exercise. $$X=\frac{\cot x+\cot y}{1+\cot x\cot y}$$ - For $\cot x$ and $\cot y$, we can apply the following identity: $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ $$X=\frac{\frac{\cos x}{\sin x}+\frac{\cos y}{\sin y}}{1+\frac{\cos x}{\sin x}\frac{\cos y}{\sin y}}$$ $$X=\frac{\frac{\cos x\sin y+\sin x\cos y}{\sin x\sin y}}{1+\frac{\cos x\cos y}{\sin x\sin y}}$$ $$X=\frac{\frac{\cos x\sin y+\sin x\cos y}{\sin x\sin y}}{\frac{\sin x\sin y+\cos x\cos y}{\sin x\sin y}}$$ $$X=\frac{\sin x\cos y+\cos x\sin y}{\cos x\cos y+\sin x\sin y}$$ Now recall that $$\sin(x+y)=\sin x\cos y+\cos x\sin y$$ $$\cos(x-y)=\cos x\cos y+\sin x\sin y$$ Therefore, $$X=\frac{\sin(x+y)}{\cos(x-y)}$$ Thus, $$\frac{\sin(x+y)}{\cos(x-y)}=\frac{\cot x+\cot y}{1+\cot x\cot y}$$ As 2 sides are equal, the equation is an identity.
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