Correct Answer - Option 2 : (a + 1)/(a - 1)
Given:
tanx/tany = a
Formula used:
sin (x + y) = sinx.cosy + cosx.siny
sin (x - y) = sinx.cosy - cosx.siny
tan θ = sin θ/cos θ
Calculation:
∵ tanx/tany = a
⇒ sinx.cosy/cosx.siny = a
⇒ sinx.cosy = a × (cosx.siny) ------(1)
∵ sin (x + y)/sin (x - y) = (sinx.cosy + cosx.siny)/(sinx.cosy - cosx.siny)
⇒ [a × (cosx.siny) + cosx.siny]/[a × (cosx.siny) - cosx.siny]
⇒ [cosx.siny(a + 1)]/[cos.siny(a - 1)]
⇒ (a + 1)/(a - 1)