prove-that-p-N-it-exist-one-polynomial-Q-2p-sin-2p-1-sin-2p-1-Q-2p-cotan-and-degQ-2p-2p-2-prove-that-k-1-p-tan-kpi-2p-1-2p-1-

Tinku Tara June 4, 2023 Algebra 0 Comments

Question Number 30742 by abdo imad last updated on 25/Feb/18

prove that ∀p ∈N it exist one polynomial Q_(2p) / sin(2p+1)θ=sin^(2p+1) θ Q_(2p) (cotanθ) and degQ_(2p) =2p 2) prove that Π_(k=1) ^p tan(((kπ)/(2p+1)))=(√(2p+1)) .

$${prove}\:{that}\:\forall{p}\:\in{N}\:\:{it}\:{exist}\:{one}\:{polynomial}\:{Q}_{\mathrm{2}{p}} \:/ \\ $$$${sin}\left(\mathrm{2}{p}+\mathrm{1}\right)\theta={sin}^{\mathrm{2}{p}+\mathrm{1}} \theta\:{Q}_{\mathrm{2}{p}} \:\left({cotan}\theta\right)\:{and}\:{degQ}_{\mathrm{2}{p}} =\mathrm{2}{p} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:{tan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{p}+\mathrm{1}}\:. \\ $$$$ \\ $$

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prove-that-p-N-it-exist-one-polynomial-Q-2p-sin-2p-1-sin-2p-1-Q-2p-cotan-and-degQ-2p-2p-2-prove-that-k-1-p-tan-kpi-2p-1-2p-1-

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