The quantity
\[\frac{\tan \frac{\pi}{5} + i}{\tan \frac{\pi}{5} - i}\]is a tenth root of unity.  In other words, it is equal to $\cos \frac{2n \pi}{10} + i \sin \frac{2n \pi}{10}$ for some integer $n$ between 0 and 9 inclusive.  Which value of $n$?
You're right, this equals $\exp{i\frac{2\pi}{5}}$. That's better. Multiplying by $-i$ yields $\frac{-i\sin\frac{\pi}{5}+\cos\frac{\pi}{5}}{-i\sin\frac{\pi}{5}-\cos\frac{\pi}{5}}$. Finally, this equals $-e^{-i\frac{2\pi}{5}}$. This equals $e^{i\frac{\pi}{10}*(5-4)}$. Since $5-4=1$, this equals $e^{i\frac{\pi}{10}*1}$. Yes, this equals $e^{i\frac{\pi}{10}}$. In other words, it equals $\cos\frac{\pi}{10}+i\sin\frac{\pi}{10}$. Yes, this equals $e^{i\frac{2\pi}{10}*3}$. This equals $\cos\frac{3\pi}{5}+i\sin\frac{3\pi}{5}$. 3 is our answer.

