$\displaystyle sin(x+y) = m$ and $\displaystyle sin(x-y) = n$, $\displaystyle \frac{tgx}{tgy}$ is ?
note that $\displaystyle \frac {\tan x}{\tan y} = \frac {\sin x \cos y}{\sin y \cos x}$
now, use the addition formula to expand $\displaystyle \sin (x + y)$ and $\displaystyle \sin (x - y)$ and see if you can form an expression equivalent to the one above