tanx + tany/ cotx + coty = (tanx)(tany)

(tanx + tany)/(cotx + coty) =
(tanx)(tany)

We will start from the left
side.

We know that:

tanx =
sinx/cosx 

cot(x) =
cosx/sin(x)

==> (tanx+tany)/(cot(x)+cot(y)) =
(sin(x)/cosx + siny/cosy)/(cosx/sinx + cosy/siny).

 =[ (
sinx*cosy + siny*cosx)/cosx*cosy]/ (cosx*siny+sinx*cosy)/
sinx*siny)

    Now we will reduce like
terms.

==>  (1/cosx*cosy) /
(1/sinx*siny)

==> sinx*siny/
cosx*cosy

==> sinx/cosx * siny / cosy = tanx * tany
..............q.e.d