Evaluating $\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$ 
How to evaluate this?
$$\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$$

My work
I couldn't figure out a method to solve this product.
I thought that this identity could help.
$$\frac{e^{i\theta}-1}{e^{i\theta}+1}=i\tan \frac{\theta}{2}$$
By supposing $z=e^{\frac{2k\pi}{2m+1}i}$, then,
$$i\tan \frac{k\pi}{2m+1}=\frac{z-1}{z+1}$$
So, the product
$$\displaystyle\prod_{k=1}^m \tan\frac{k\pi}{2m+1}=\displaystyle\prod_{k=1}^m \frac{z-1}{i(z+1)}=\displaystyle\prod_{k=1}^m \frac{e^{\frac{2k\pi}{2m+1}i}-1}{i(e^{\frac{2k\pi}{2m+1}i}+1)}$$
which is getting more complicated.
Answer is $\sqrt{2m+1}$. Any help is appreciated.

So you can start by breaking the product to get:
$$\prod_{k=1}^{2m}\tan{(\frac{k\pi}{2m+1})}=\prod_{k=1}^{2m}\sin{(\frac{k\pi}{2m+1})}\prod_{k=1}^{2m}\frac{1}{\cos{(\frac{k\pi}{2m+1}})}$$
Now :
$$\prod_{k=1}^{2m}\cos{(\frac{k\pi}{2m+1})}=\frac{(-1)^{m}}{2^{2m}}$$
Proof:
$$\cos{(\frac{k\pi}{2m+1})}=\exp{(\frac{ki\pi}{2m+1})}\frac{(1+\exp{(\frac{-2ik\pi}{2m+1}}))}{2}$$
If we have polynomial $z^{2m+1}-1$ then its $2m+1$ roots of unity are $\exp{(\frac{-2ik\pi}{2m+1}}):1\le k \le 2m+1$
Thus :
$$z^{2m+1}-1=-\prod_{k=1}^{2m+1}(-z+\exp{(\frac{-2ik\pi}{2m+1}}))$$
$$\therefore \frac{(-1)^{2m+1}-1}{2}=-1=-\prod_{k=1}^{2m}(1+\exp{(\frac{-2ik\pi}{2m+1}}))$$
$$\therefore \prod_{k=1}^{2m}\cos{(\frac{k\pi}{2m+1})}=\frac{1}{2^{2m}}\exp(\frac{i(2m)(2m+1)\pi}{2(2m+1)})=\frac{(-1)^{m}}{2^{2m}}$$
Now we have to evaluate : $$\prod_{k=1}^{2m}\sin{(\frac{k\pi}{2m+1})}$$
Using similar process like one above we get:
$$\prod_{k=1}^{2m}\sin{(\frac{k\pi}{2m+1})}=(-1)^{m}\lim_{z\to1}\frac{z^{2m+1}-1}{(z-1)(2i)^{2m+1}}=\frac{(2m+1)(-1)^{m}}{(2i)^{2m}}$$
Okay we have these two products but how do they relate to the original problem. Well
$$\tan{(\frac{(2m+1-k)\pi}{2m+1})}=\tan{(\pi-\frac{k\pi}{2m+1})}=-\tan{(\frac{k\pi}{2m+1})}$$
thus:
$$\prod_{k=1}^{2m}\tan{(\frac{k\pi}{2m+1})}=\prod_{k=1}^{m}\tan{(\frac{k\pi}{2m+1})}\prod_{k=m+1}^{2m}\tan{(\frac{k\pi}{2m+1})}=(-1)^{m}(\prod_{k=1}^{m}\tan{(\frac{k\pi}{2m+1})})^{2}$$
$$\prod_{k=1}^{2m}\tan{(\frac{k\pi}{2m+1})}=\frac{(2m+1)}{(2i)^{2m}}2^{2m}=(-1)^{m}(2m+1)$$
$$\therefore (-1)^{m}(\prod_{k=1}^{m}\tan{(\frac{k\pi}{2m+1})})^{2}=(-1)^{m}(2m+1)$$
$$\therefore \prod_{k=1}^{m}\tan{(\frac{k\pi}{2m+1})}=\sqrt{2m+1} $$
