Value of $\tan^2(5)+\tan^2(10)+\tan^2(15)+...+\tan^2(85)$ Calculate the exact value of $$\tan^2(5^\circ)+\tan^2(10^\circ)+\tan^2(15^\circ)+\cdots+\tan^2(85^\circ)$$
How to evaluate this sum of all these values? Is there a specific way? Thanks in advance. 

$$\tan^2(5^\circ)+\tan^2(10^\circ)+...+\tan^2(85^\circ) = \tan^2\bigg(\frac{5\pi}{180}\bigg) +\tan^2\bigg(\frac{10\pi}{180}\bigg)+...+\tan^2\bigg(\frac{85\pi}{180}\bigg)$$
$$= \sum_{r=1}^{17}\tan^2\bigg(\frac{r\pi}{2\cdot18}\bigg)$$
Now note this result: Prove that $\sum\limits_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$
We have $n = 18$ so then we get:
$$S = \frac{(18-1)(2\cdot 18 -1)}{3} =\frac{595}{3}$$
