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

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(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

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