If x, y, z are in A.P. and tan x, tan y, and tan z are also in A.P., then

x, y and z are in A.P.
So, 2y = x + z.. (1)
and tan x, tan y and tan z are in A.P.
So 2tan y = tan x + tan z
2tan y = tan[x + z/(1 - xz)]
tan(2tan y) = x + z/(1 - xz)
Formula [tan 2θ = 2tan θ/(1 - tan²θ)]
2tan(tan y)/(1 - [tan(tan y)]²) = x + z/(1 - xz)
2y/(1 - y²) = x + z/(1 - xz)
From eqn. (1) 2y = x + z
(x + z)/(1 - y²) = x + z/(1 - xz)
(x + z)[1/(1 - y²) - 1/(1 - xz)] = 0
x + z = 0 or 1/(1 - y²) = 1/(1 - xz)
1 - y² = 1 - xz
y² = xz
So, x, y and z are in G.P.