alfa+bita = thita & tan x ratio tan y= x ratio y then prove that sin alfa- bita = x- y ratio x+y sinthita

Answer 1

that's alpha, beta, theta

If I read you right, we have

θ = α+β
tanx/tany = x/y
prove: sin(α-β) = (x-y)/(x+y) sinθ

well, sinθ = sin(α+β)
so now we can say we want

sin(α-β)/sin(α+β) = (x-y)/(x+y)

note that

sin(x-y)+sin(x+y) = 2sinx*cosy
sin(x-y)-sin(x+y) = -2cosx*siny

Things should work out from here