Solution: We need to show that the function
\[
\delta (x,y) = \left\vert \tan ^{-1} x - \tan ^{-1} y \right\vert
\]
is a metric on $\mathbb{R} $. We will verify each property of a metric.
-
$\delta (x,y) \geq 0$ for any $x,y \in \mathbb{R} $.
This is clear from the definition.
-
$\delta (x,y) = 0$ if and only if $x = y$.
Let
\begin{align*}
\delta (\times y) = 0 & \implies \left\vert \tan ^{-1} x - \tan ^{-1} y \right\vert = 0 \\
& \implies \tan ^{-1} x - \tan ^{-1} y = 0 \\
& \implies \tan ^{-1} x = \tan ^{-1} y \\
& \implies x = y,
\end{align*}
where the last implication is true because $\tan ^{-1} $ is an injective function.
-
$\delta (x,y) = \delta (y,x)$.
This is also clear as
\begin{align*}
\delta (x,y) & = \left\vert \tan ^{-1} x - \tan ^{-1} y \right\vert \\
& = \left\vert \tan ^{-1} y - \tan ^{-1} x \right\vert \\
& = \delta (y,x).
\end{align*}
-
$\delta (x,z) \leq \delta (x,y) + \delta (y,z)$ for any $x,y,z$.
Consider
\begin{align*}
\delta (x,z) & = \left\vert \tan ^{-1} x - \tan ^{-1} y \right\vert\\
& = \left\vert \tan ^{-1} x \textcolor{red}{- \tan ^{-1} y + \tan ^{-1} y} - \tan ^{-1} z \right\vert \\
& \leq \left\vert \tan ^{-1} x - \tan ^{-1} y \right\vert + \left\vert \tan ^{-1} y - \tan ^{-1} z \right\vert \\
& = \delta (x,y) + \delta (y,z).
\end{align*}
Thus, we proved that $\delta $ is a metric on $\mathbb{R} $.