I believe we can show that
$\prod_{m=1}^{2k} \cot \frac{m\pi}{2k+1} = (-1)^k \frac{1}{2k+1}$
Using Chebyshev Polynomials $\displaystyle T_n(x)$ (of degree $\displaystyle n$, and leading coefficient $\displaystyle 2^{n-1}$), which satisfy
$T_{n}(\cos x) = \cos (n x)$
and
$T_{2n+1}(\sin x) = (-1)^k\sin ((2n+1) x)$
We get the product of roots of $\displaystyle T_{2k+1}(x) = -1$, to find $\displaystyle \prod_{m=1}^{2k+1} (-1)^k\cos \frac{m\pi}{2k+1}$
(Note that its roots are $\displaystyle \cos (\frac{(2r+1)\pi}{2k+1})$ and $\displaystyle -\cos (\frac{2r\pi}{2k+1})$)
which gives us $\prod_{m=1}^{2k} \cos \frac{m\pi}{2k+1} = \frac{(-1)^k}{2^{2k}}$
To find $\displaystyle \prod_{m=1}^{2k} \sin \frac{m\pi}{2k+1}$, we need to find the product of roots of $\displaystyle \frac{T_{2k+1}(x)}{x} = 0$.
We can prove that the coefficient of $\displaystyle x$ in $\displaystyle T_{2k+1}(x)$ is $\displaystyle 2k+1$ and that would give us
$\prod_{m=1}^{2k} \sin\frac{m\pi}{2k+1} = \frac{2k+1}{2^{2k}}$
Thus
$\prod_{m=1}^{2k} \cot\frac{m\pi}{2k+1} = \frac{(-1)^k}{2k+1}$
Since $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = \int_{0}^{1} \frac{1}{1+x^2} \text{ dx} = \frac{\pi}{4}$
we are done.