let-give-A-cos-sin-sin-cos-1-calculate-t-A-A-prove-that-A-is-inversible-and-find-A-1-2-find-A-n-for-n-N-3-developp

Question Number 28259 by abdo imad last updated on 22/Jan/18

l e t g i v e A = (c o s θ - s i n θ)

(s i n θ c o s θ)

1) c a l c u l a t e^{t} A . A . p r o v e t h a t A i s i n v e r s i b l e a n d f i n d

A^{- 1}

2) f i n d A^{n} f o r n \in N

3) d e v e l o p p {(A + A^{- 1})}^{n} t h e n p r o v e t h a t

2^{n} {c o s}^{n} θ = \sum_{k = 0}^{n} C_{n}^{k} (n - 2 k) θ a n d

\sum_{k = 0}^{n} C_{n}^{n} s i n (n - 2 k) θ = 0 .