Prove that the roots of the equation \( 8 x^{3}-4 x^{2}-4 x+1=0 \) are \( \cos \frac{\pi}{7}, \c...
Prove that the roots of the equation \( 8 x^{3}-4 x^{2}-4 x+1=0 \) are \( \cos \frac{\pi}{7}, \cos \frac{3 \pi}{7}, \cos \frac{5 \pi}{7} \) 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 \( \tan ^{2} \frac{\pi}{7}, \tan ^{2} \frac{3 \pi}{7}, \tan ^{2} \frac{5 \pi}{7} \).
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