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Question Number 28265 by abdo imad last updated on 22/Jan/18
1)  find P∈R[x] / P(sinx) =sin(2n+1)x  2) find the roots of P and degP  3) decompose  (1/P)  and prove that  ((2n+1)/(sin(2n+1)x)) = Σ_(k=0) ^(2n)     (((−1)^k  cos(((kπ)/(2n+1))))/(sinx−sin (((kπ)/(2n+1))))))  .
$$\left.\mathrm{1}\right)\:\:{find}\:{P}\in{R}\left[{x}\right]\:/\:{P}\left({sinx}\right)\:={sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\:{and}\:{degP} \\ $$$$\left.\mathrm{3}\right)\:{decompose}\:\:\frac{\mathrm{1}}{{P}}\:\:{and}\:{prove}\:{that} \\ $$$$\frac{\mathrm{2}{n}+\mathrm{1}}{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x}}\:=\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)}{{sinx}−{sin}\:\left(\frac{{k}\pi}{\left.\mathrm{2}{n}+\mathrm{1}\right)}\right)}\:\:. \\ $$

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