$Prove that$


\tan \theta + \tan \left (\theta + \frac{\pi}{n} \right ) + \tan \left (\theta + \frac{2\pi}{n} \right ) + \dots + \tan \left (\theta + \frac{(n-1)\pi}{n} \right ) = -n \cot \left (\frac{n\pi}{2} + n\theta \right )

Induction doesn't work, obviously. How can I do it? Any help is very much appreciated!