Question Number 30598 by abdo imad last updated on 23/Feb/18
$${prove}\:{that}\:{it}\:{exist}\:{one}\:{polynomial}\:{p}/ \\ $$$${p}\left({cosx}\right)={cos}\left({nx}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:. \\ $$
Commented by abdo imad last updated on 27/Feb/18
$${we}\:{have}\:{by}\:{moivre}\:{formula}\: \\ $$$${cos}\left({nx}\right)\:+{isin}\left({nx}\right)=\left({cosx}\:+{isinx}\right)^{{n}} \\ $$$$=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:\left({isinx}\right)^{{k}} \:\left({cosx}\right)^{{n}−{k}} \\ $$$$=\:\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:{C}_{{n}} ^{\mathrm{2}{p}} \:\left({isinx}\right)^{\mathrm{2}{p}} \:\left({cosx}\right)^{{n}−\mathrm{2}{p}} \:+\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}−\mathrm{1}}{\mathrm{2}}\right]} =\:{C}_{{n}} ^{\mathrm{2}{p}+\mathrm{1}} \left({isinx}\right)^{\mathrm{2}{p}+\mathrm{1}} \left({cosx}\right)^{{n}−\mathrm{2}{p}−\mathrm{1}} \\ $$$${cos}\left({nx}\right)={Re}\left({e}^{{inx}} \right)=\:\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}} ^{\mathrm{2}{p}} \:\left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right)^{{p}} \left({cosx}\right)^{{n}−\mathrm{2}{p}} \\ $$$$={p}\left({cosx}\right)\:/{p}\left({x}\right)=\:\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}} ^{\mathrm{2}{p}} \:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{p}} \:{x}^{{n}−\mathrm{2}{p}} \:. \\ $$$$\left.\mathrm{2}\right)\:{X}\:{root}\:{of}\:{p}\:\left({x}\right)\Leftrightarrow\:{p}\left({X}\right)=\mathrm{0}\:\:{let}\:{put}\:{X}={cos}\theta \\ $$$${p}\left({X}\right)=\mathrm{0}\:\Leftrightarrow{p}\left({cos}\theta\right)=\mathrm{0}\:\Leftrightarrow{cos}\left({n}\theta\right)=\mathrm{0} \\ $$$$\Leftrightarrow\:\:{n}\theta=\:\frac{\pi}{\mathrm{2}}\:+{k}\pi\:\:\Leftrightarrow\:\theta=\frac{\pi}{\mathrm{2}{n}}\:+\frac{{k}\pi}{{n}}=\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{\mathrm{2}{n}}\:{but}\:{wecan}\:{chow} \\ $$$${that}\:{deg}\:{p}={n}\:\Rightarrow\:{the}\:{roits}\:{of}\:{p}\left({x}\right)\:{are} \\ $$$${X}_{{k}} =\:{cos}\left(\theta_{{k}} \right)\:={cos}\left(\left(\mathrm{2}{k}+\mathrm{1}\right)\frac{\pi}{\mathrm{2}{n}}\right)\:{with}\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]. \\ $$