If

$\frac{\cos (x-y)}{\cos (x+y)}=\frac{m}{n}$

$\Rightarrow \frac{\cos x \cos y+\sin x \sin y}{\cos x \cos y-\sin x \sin y}=\frac{m}{n}$

$\Rightarrow \frac{1+\tan x \tan y}{1-\tan x \tan y}=\frac{m}{n}$   [ Dividing numerator and denominator of LHS by $\cos x \cos y$ ]

$\Rightarrow n+n \tan x \tan y=m-m \tan x \tan y$

$\Rightarrow \tan x \tan y(m+n)=m-n$

$\Rightarrow \tan x \tan y=\frac{m-n}{m+n}$