KöMaL 2011 B4371

Prove that $\frac{1}{\sin^2{\frac{\pi}{14}}} + \frac{1}{\sin^2{\frac{3\pi}{14}}} + \frac{1}{\sin^2{\frac{5\pi}{14}}} = 24.$ (Suggested by B. Kovács, Szatmárnémeti)