Mathematics

Problem

Prove that the roots of the equation

8x^3 - 4x^2 - 4x + 1 = 0 \text{ are } \cos \frac{\pi}{7}, \cos \frac{3\pi}{7}, \cos \frac{5\pi}{7} \text{ and}

hence, show that

\sec \frac{\pi}{7} + \sec \frac{3\pi}{7} + \sec \frac{5\pi}{7} = 4

and deduce the equation whose roots are

\tan^2 \frac{\pi}{7}, \tan^2 \frac{3\pi}{7}, \tan^2 \frac{5\pi}{7}