KöMaL 2011 A546

Show that $\frac{1}{\sin^2{\dfrac{\pi}{4k+2}}} + \frac{1}{\sin^2{\dfrac{3\pi}{4k+2}}} + \frac{1}{\sin^2{\dfrac{5\pi}{4k+2}}} + \ldots + \frac{1}{\sin^2{\dfrac{(2k-1)\pi}{4k+2}}} = 2k(k+1)$ for every positive integer $k$.