Question Number 30742 by abdo imad last updated on 25/Feb/18
$${prove}\:{that}\:\forall{p}\:\in{N}\:\:{it}\:{exist}\:{one}\:{polynomial}\:{Q}_{\mathrm{2}{p}} \:/ \\ $$$${sin}\left(\mathrm{2}{p}+\mathrm{1}\right)\theta={sin}^{\mathrm{2}{p}+\mathrm{1}} \theta\:{Q}_{\mathrm{2}{p}} \:\left({cotan}\theta\right)\:{and}\:{degQ}_{\mathrm{2}{p}} =\mathrm{2}{p} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:{tan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{p}+\mathrm{1}}\:. \\ $$$$ \\ $$