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)))))) .


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Question Number 28265 by abdo imad last updated on 22/Jan/18

$1) f i n d P \in R [x] / P (s i n x) = s i n (2 n + 1) x$ $2) f i n d t h e r o o t s o f P a n d d e g P$ $3) d e c o m p o s e \frac{1}{P} a n d p r o v e t h a t$ $\frac{2 n + 1}{s i n (2 n + 1) x} = \sum_{k = 0}^{2 n} \frac{{(- 1)}^{k} c o s (\frac{k π}{2 n + 1})}{s i n x - s i n (\frac{k π}{2 n + 1)})} .$
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