In general, By DeMoivre's Theorem, \begin{align*} \operatorname{cis} n \theta &= (\operatorname{cis} \theta)^n \\ &= (\cos \theta + i \sin \theta)^n \\ &= \cos^n \theta + \binom{n}{1} i \cos^{n - 1} \theta \sin \theta - \binom{n}{2} \cos^{n - 2} \theta \sin^2 \theta - \binom{n}{3} i \cos^{n - 3} \theta \sin^3 \theta + \dotsb.\end{align*}Matching real and imaginary parts, we get \begin{align*} \cos n \theta &= \cos^n \theta - \binom{n}{2} \cos^{n - 2} \theta \sin^2 \theta + \binom{n}{4} \cos^{n - 4} \theta \sin^4 \theta - \dotsb, \\ \sin n \theta &= \binom{n}{1} \cos^{n - 1} \theta \sin \theta - \binom{n}{3} \cos^{n - 3} \theta \sin^3 \theta + \binom{n}{5} \cos^{n - 5} \theta \sin^5 \theta - \dotsb.\end{align*}Therefore, \begin{align*} \tan n \theta &= \frac{\sin n \theta}{\cos n \theta} \\ &= \frac{\dbinom{n}{1} \cos^{n - 1} \theta \sin \theta - \dbinom{n}{3} \cos^{n - 3} \theta \sin^3 \theta + \dbinom{n}{5} \cos^{n - 5} \theta \sin^5 \theta - \dotsb}{\cos^n \theta - \dbinom{n}{2} \cos^{n - 2} \theta \sin^2 \theta + \dbinom{n}{4} \cos^{n - 4} \theta \sin^4 \theta - \dotsb} \\ &= \frac{\dbinom{n}{1} \tan \theta - \dbinom{n}{3} \tan^3 \theta + \dbinom{n}{5} \tan^5 \theta - \dotsb}{1 - \dbinom{n}{2} \tan^2 \theta + \dbinom{n}{4} \tan^4 \theta - \dotsb}.\end{align*}Taking $ n = 9, $ we get \[\tan 9 \theta = \frac{9 \tan \theta - 84 \tan^3 \theta + 126 \tan^5 \theta - 36 \tan^7 \theta + \tan^9 \theta}{1 - 36 \tan^2 \theta + 126 \tan^4 \theta - 84 \tan^6 \theta + 9 \tan^8 \theta}.\]Note that for $ \theta = 5^\circ, $ $ 25^\circ, $ $ \dots, $ $ 165^\circ, $ $ \tan 9 \theta = \tan 45^\circ = 1 $. Thus, \[1 = \frac{9 \tan \theta - 84 \tan^3 \theta + 126 \tan^5 \theta - 36 \tan^7 \theta + \tan^9 \theta}{1 - 36 \tan^2 \theta + 126 \tan^4 \theta - 84 \tan^6 \theta + 9 \tan^8 \theta}.\]Let $ t = \tan \theta, $ so \[1 = \frac{9t - 84t^3 + 126t^5 - 36t^7 + t^9}{1 - 36t^2 + 126t^4 - 84t^6 + 9t^8}.\]Thus, $ \tan 5^\circ, $ $ \tan 25^\circ, $ $ \dots, $ $ \tan 165^\circ $ are the roots of \[t^9 - 9t^8 - 36t^7 + 84t^6 + 126t^5 - 126t^4 - 84t^3 + 36t^2 + 9t - 1 = 0.\]By Vieta's formulas, their sum is $ \boxed{9} $.