#### Answer

sample answer: $\delta=0.01$

#### Work Step by Step

Graph (see below)
y=$\tan$x,
$y=1\pm 0.2$
(parallel lines, 0.2 above and below y=1 ,
to outline the interval on the y-axis
to which the function values belong),
$x=\arctan 0.8,\quad x=\arctan 1.2$
vertical lines to outline the interval
$I=(0.675, 0.876)$ around x=$\displaystyle \frac{\pi}{4}$.
From the graph, $x\in I \ \ \Rightarrow\ \ |f(x)-1| < 0.2$.
(all values of $x\in I$ have f(x) within $\pm 0.2 $ of $1$)
We take a subset from $I$,
one that we can write as$ (\displaystyle \frac{\pi}{4}-\delta,\frac{\pi}{4}+\delta)$,
such as, for example, for $\delta=0.01:$
$I_{1}=(\displaystyle \frac{\pi}{4}-0.01, \frac{\pi}{4}+0.01)$
(the problem asks to find a $\delta$, not the greatest possible $\delta)$
This interval is such that,
for $x\in I_{1}\ \ \Rightarrow\ \ x\in I\ \ \Rightarrow\ \ |f(x)-1| < 0.2$.
that is,
$ 0 < |x-\displaystyle \frac{\pi}{4}| < \delta$, then $|\tan \mathrm{x}-1| < \epsilon$.