Answer
$\tan x-\tan y=\frac{\sin(x-y)}{\cos x\cos y}$
Work Step by Step
Start with the right side:
$\frac{\sin(x-y)}{\cos x\cos y}$
Expand using the subtraction formula for sine:
$=\frac{\sin x\cos y-\cos x\sin y}{\cos x\cos y}$
Simplify:
$=\frac{\sin x\cos y}{\cos x\cos y}-\frac{\cos x\sin y}{\cos x\cos y}$
$=\frac{\sin x}{\cos x}-\frac{\sin y}{\cos y}$
$=\tan x-\tan y$
Since this equals the left side, the identity has been proven.
