Prove that:

(sin(x + y) / sin(x - y)) = (tan x + tan y) / (tan x - tan y)

L.H.S. 

=> sin(x + y) / sin(x - y) = (sin x cos y + cos x sin y) / (sin x cos y - cos x sin y)

Dividing numerator and denominator by cos x cos y,

 we get:

=> sin(x + y) / sin(x - y) = (tan x + tan y) / (tan x - tan y)