# Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 228: 67

$$\frac{\sin(x-y)}{\sin(x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}$$ The equation has been verified to be an identity.

#### Work Step by Step

$$\frac{\sin(x-y)}{\sin(x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}$$ About the right side, we can use the identity $\tan x=\frac{\sin x}{\cos x}$ $$\frac{\tan x-\tan y}{\tan x+\tan y}$$ $$=\frac{\frac{\sin x}{\cos x}-\frac{\sin y}{\cos y}}{\frac{\sin x}{\cos x}+\frac{\sin y}{\cos y}}$$ $$=\frac{\frac{\sin x\cos y-\cos x\sin y}{\cos x\cos y}}{\frac{\sin x\cos y+\cos x\sin y}{\cos x\cos y}}$$ $$=\frac{\sin x\cos y-\cos x\sin y}{\sin x\cos y+\cos x\sin y}$$ $$=\frac{\sin (x-y)}{\sin(x+y)}$$ (identities of sine of a sum and a difference) Therefore, the equation is an identity.

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