\(\begin{aligned} \frac{\sin (x-y)}{\sin (x+y)} &=\frac{\tan x-\tan y}{\tan x+\tan y} \\ \frac{\sin x \cos y-\cos x \sin y}{\sin x \cos y+\cos x \sin y} &=\\ \frac{\sin x \cos y-\cos x \sin y}{\sin x ...\(\begin{aligned} \frac{\sin (x-y)}{\sin (x+y)} &=\frac{\tan x-\tan y}{\tan x+\tan y} \\ \frac{\sin x \cos y-\cos x \sin y}{\sin x \cos y+\cos x \sin y} &=\\ \frac{\sin x \cos y-\cos x \sin y}{\sin x \cos y+\cos x \sin y} \cdot \frac{\left(\frac{1}{\cos x \cdot \cos y}\right)}{\left(\frac{1}{\cos x \cdot \cos y}\right)} &=\\ \frac{\left(\frac{\sin x}{\cos x \cdot \cos y}\right)-\left(\frac{\cos x \sin y}{\cos x \cdot \cos y}\right)}{\left(\frac{\sin x \cos x}{\cos x \cdot \cos y}\right)+\left(\…