If sin( y+z -x), sin(z + x-y), sin(x + y - z) are in A.P, then tanx, tany, tanz are in

 Applying b - a= c - b for A.P., 

we get 2 cosz sin(x - y) - 2 cosx sin(y - z) 

Dividing by 2 cosx cosy cosz, etc., 

we get tanx-tany =tany -tanz