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(tan x)/(tan y) = [(sin x)/(cos x)]/[sin y)/(cos y) = [(sin x)(cos y)]/[(cos x)(sin y)]
[sin (x+y)]/[sin (x - y)] = (a + b)/(a - b)
sin (x + y) = a + b
(sin x)(cos y) + (cos x)(sin y) = a + b
(sin x)(cos y) = a
(cos x)(sin y) = b
(tan x)/(tan y) = a/b
If sin(x+y)/sin(x-y)=(a+b)/(a-b) then tanx/tany=?
sin(x+y)/sin(x-y)=(sinxcosy+cosxsiny)/(sinxcosy-cosxsiny)  divide both numerator and denominator by
cosxcosy and get (tanx+tany)/(tanx-tany)
Let a=tanx, b=tany and have sin(x+y)/sin(x-y)=(a+b)/(a-b) and also tanx/tany=a/b