If `(sin(x+y))/(sin(x-y))=(a+b)/(a-b)` , then show that `(tanx)/(tany)=(a)/(b)`.

Explanation

Given that, `(sin(x+y))/(sin(x-y))=(a+b)/(a-b)` ltBrgt Using componendo and dividendo, ltBrgt `rArr" "=(sin(x +y)+ [sin(x-y)])/(sin(x+y) -sin(x-y))=(a+b+a-b)/(a+b-a+b )` ltBrgt `rArr" "=(2sin((x+y+x-y)/(2))*cos((x+y-x+y)/(2)))/(2cos((x+y+x-y)/(2))*sin((x+y-x+y)/(2)))=(2a)/(2b)`
`[because sinx+siny=2sin""(x+y)/(2)*cos""(x+y)/(2) and sinx-siny=2cos""(x+y)/(2)*sin""(x-y)/(2)]`
`rArr " "=(sinx*cosy)/(cosx*siny)=(a)/(b) `
`rArr" "(tanx)/( tany)=(a)/(b)`