Answer
$\frac{\sqrt{3}}{3}$
Work Step by Step
We use the subtraction formula for cosine, $\tan(s+t)=\frac{\tan s+\tan t}{1-\tan s \tan t}$, with $s=\frac{\pi}{18}$ and $t=\frac{\pi}{9}$.
$\frac{\tan \frac{\pi}{18}+\tan \frac{\pi}{9}}{1-\tan \frac{\pi}{18} \tan \frac{\pi}{9}}$
$=\tan(\frac{\pi}{18}+\frac{\pi}{9})$
$=\tan(\frac{\pi}{18}+\frac{2\pi}{18})$
$=\tan(\frac{3\pi}{18})$
$=\tan(\frac{\pi}{6})$
$=\frac{\sqrt{3}}{3}$
