Answer
$\delta = 0.09$
Work Step by Step
On the graph, we can see that when $0.675 \lt x \lt 0.876$, then $0.8 \lt tan ~x \lt 1.2$
$\vert \frac{\pi}{4}-0.675\vert = 0.11$
$\vert \frac{\pi}{4}-0.876 \vert= 0.09$
Thus, when $\vert x-\frac{\pi}{4} \vert \lt 0.09$, then $\vert tan ~x - 1\vert \lt 0.2$
Therefore, we can let $\delta = 0.09$
