Complex Complex Equations

Algebra Level pending

Given \(z=\cos(\frac{2\pi}{2n+1})+i\sin(\frac{2\pi}{2n+1})\),where n is a positive integer,find the equation whose roots are \(\alpha=z+z^3+z^5+...+z^{2n-1}\) and \(\beta=z^2+z^4+z^6+...+z^{2n}\).

The equation will be of form: \[x^2+x+\frac{1}{k}sec^2(\frac{\pi}{2n+1})\]What is k?

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