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


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Question Number 29837 by abdo imad last updated on 12/Feb/18

$l e t g i v e T_{n} (x) = c o s (n a r c o s x) w i t h x \in [- 1, 1]$ $1) p r o v e t h a t T_{n} i s a p o l y n o m i a l a n d T_{n} \in Z [x]$ $2) c a l c u l a t e T_{1}, T_{2}, T_{3}, a n d T_{4}$ $3) p r o v e t h a t T_{n + 2} (x) = 2 x T_{n + 1} (x) - T_{n} (x)$ $4) f i n d t h e r o o t s o f T_{n} a n d f a c t o r i z e T_{n} (x) .$
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