Explanation

(a)
Since, $x,\ y,\ z$ are in AP
$\therefore\ \ \ \ \ \ \ \ \ \ \ y = \frac{x + z}{2}$ …(i)
And $\tan^{- 1}{x,\tan^{- 1}y}$ and $\tan^{- 1}z$ are also in AP.
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\tan^{- 1}{y = \tan^{- 1}{x + \tan^{- 1}z}}$
$\Rightarrow \ \tan^{- 1}\left( \frac{2y}{1 - y^{2}} \right) = \tan^{- 1}\left( \frac{x + z}{1 - xz} \right)$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2y}{1 - y^{2}} = \frac{2y}{1 - xz}$ [from Eq. (i)]
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y^{2} = xz$
$\Rightarrow \ \ x,\ y,\ z$ are in GP.
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ x = y = z$