Answer
$-1 \lt tan ~x \lt 1~~$ when $~~x~~$ is in:
$[0,\frac{\pi}{4})\cup (\frac{3\pi}{4},\frac{5\pi}{4}) \cup (\frac{7\pi}{4},2\pi]$
Work Step by Step
Note that $tan~x = 1$ when $x= \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$
Note that $tan~x = -1$ when $x= \frac{3\pi}{4}$ and $x = \frac{7\pi}{4}$
We can use the graph of $tan~x$ to solve this question:
On the interval $[0,\frac{\pi}{4})$:
$0 \leq tan ~x \lt 1$
On the interval $(\frac{3\pi}{4},\frac{5\pi}{4})$:
$-1 \lt tan ~x \lt 1$
On the interval $(\frac{7\pi}{4},2\pi]$:
$-1 \lt tan ~x \leq 0$
Therefore, $~~-1 \lt tan ~x \lt 1~~$ when $~~x~~$ is in:
$[0,\frac{\pi}{4})\cup (\frac{3\pi}{4},\frac{5\pi}{4}) \cup (\frac{7\pi}{4},2\pi]$