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Question Number 96293 by 06122004 last updated on 31/May/20

Answered by mathmax by abdo last updated on 31/May/20

$$\mathrm{let}\mathrm{P}\left(\mathrm{x}\right)=(\mathrm{x}-\frac{1}{\mathrm{cosa}})(\mathrm{x}-\frac{1}{\cos \left(2\mathrm{a}\right)})....(\mathrm{x}-\frac{1}{\cos \left(\mathrm{na}\right)})=\prod _{\mathrm{k}=1}^{\mathrm{n}}(\mathrm{x}-\frac{1}{\cos \left(\mathrm{ka}\right)})$$$$\mathrm{so}\frac{1}{\cos \left(\mathrm{ka}\right)}\mathrm{are}\mathrm{roots}\mathrm{of}\mathrm{this}\mathrm{polynom}$$$$\mathrm{we}\mathrm{have}\frac{{\mathrm{P}}^{{}^{\prime }}\left(\mathrm{x}\right)}{\mathrm{P}\left(\mathrm{x}\right)}=\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\mathrm{x}-\cos \left(\mathrm{ka}\right)}$$$$\mathrm{x}=0\Rightarrow \sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{\cos \left(\mathrm{ka}\right)}=-\frac{{\mathrm{P}}^{{}^{\prime }}\left(0\right)}{\mathrm{P}\left(0\right)}$$$$\mathrm{P}\left(0\right)=\frac{{(-1)}^{\mathrm{n}}}{\mathrm{cosa}.\cos \left(2\mathrm{a}\right)....\cos \left(\mathrm{na}\right)}\mathrm{rest}\mathrm{to}\mathrm{find}{\mathrm{P}}^{{}^{\prime }}\left(0\right)....\mathrm{be}\mathrm{continued}...$$

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