$\tan$ is bounded outside the $\epsilon$-neighborhood of its poles

Question

Good time of day. Can you help me with this question. Is it true that the function $\tan$ is bounded outside the $\epsilon$-neighborhood of its poles?

Thank you

Answer 1

Hint: Consider that $\tan$ is strictly increasing and continuous on $(-\frac{\pi}{2},\frac{\pi}{2})$, so its image on $[a,b]\subset (-\frac{\pi}{2},\frac{\pi}{2})$ is $[\tan a,\tan b]$.