We have, L.H.S. = 2 cos $\frac{\pi}{13}$ cos $\frac{9\pi}{13}$ + cos $\frac{3\pi}{13}$ + cos $\frac{5\pi}{13}$ = cos $\left(\frac{9\pi}{13}+\frac{\pi}{13}\right)$ + cos $\left(\frac{9\pi}{13}-\frac{\pi}{13}\right)$ + cos $\frac{3\pi}{13}$ + cos $\frac{5\pi}{13}$ = cos $\frac{10\pi}{13}$ + cos $\frac{8\pi}{13}$ + cos $\frac{3\pi}{13}$ + cos $\frac{5\pi}{13}$ = cos $\left(\frac{13\pi-3\pi}{13}\right)$ + cos $\left(\frac{13\pi-5\pi}{13}\right)$ + cos $\frac{3\pi}{13}$ + cos $\frac{5\pi}{13}$ = - cos $\frac{3\pi}{13}$ - cos $\left(\frac{5\pi}{13}\right)$ + cos $\frac{3\pi}{13}$ + cos $\frac{5\pi}{13}$ Since cos (π - θ) = - cos θ = 0 = R.H.S.