If x, y and z are in A.P and ${\mathrm{Tan}}^{-1}\mathrm{x},{\mathrm{Tan}}^{-1}\mathrm{y}\text{ and }{\mathrm{Tan}}^{-1}\mathrm{z}$ are also in A.P then

$$\begin{gathered} 2{\mathrm{Tan}}^{-1}\mathrm{y}={\mathrm{Tan}}^{-1}\mathrm{x}+{\mathrm{Tan}}^{-1}\mathrm{z} \\ \Rightarrow {\tan }^{-1}\left(\frac{2\mathrm{y}}{1-{\mathrm{y}}^2}\right)={\tan }^{-1}\left(\frac{\mathrm{x}+\mathrm{z}}{1-\mathrm{xz}}\right) \\ \Rightarrow \frac{2\mathrm{y}}{1-{\mathrm{y}}^2}=\frac{\mathrm{x}+\mathrm{z}}{1-\mathrm{xz}}\Rightarrow \frac{2\mathrm{y}}{1-{\mathrm{y}}^2}=\frac{2\mathrm{y}}{1-\mathrm{xz}} \left(\because 2\mathrm{y}=\mathrm{x}+\mathrm{z}\right) \\ \Rightarrow 1-\mathrm{xz}=1-{\mathrm{y}}^2\Rightarrow {\mathrm{y}}^2=\mathrm{xz} \end{gathered}$$
$\therefore$ x,y,z are in G.P as well as in A.P $\Rightarrow$ x = y = z