Question Number 29837 by abdo imad last updated on 12/Feb/18
![let give T_n (x)=cos(n arcosx) with x∈[−1,1] 1) prove that T_n is a polynomial and T_n ∈Z[x] 2)calculate T_1 , T_2 , T_3 ,and T_4 3) prove that T_(n+2) (x)=2x T_(n+1) (x)−T_n (x) 4)find the roots of T_n and factorize T_n (x).](Q29837.png)
$${let}\:{give}\:\:{T}_{{n}} \left({x}\right)={cos}\left({n}\:{arcosx}\right)\:{with}\:{x}\in\left[−\mathrm{1},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{T}_{{n}} \:{is}\:{a}\:{polynomial}\:{and}\:{T}_{{n}} \in{Z}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right){calculate}\:{T}_{\mathrm{1}} ,\:{T}_{\mathrm{2}} ,\:{T}_{\mathrm{3}} ,{and}\:{T}_{\mathrm{4}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{T}_{{n}+\mathrm{2}} \left({x}\right)=\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} \left({x}\right)−{T}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{factorize}\:{T}_{{n}} \left({x}\right). \\ $$