Let be a>0. Given x_(0) > 0 we define recursively the sequence (x_(n)), which is infinite sequence starting with n=1. The recursive formula of this sequence is:

x_(n+1) = x_(n) - {cos x_(n) * sin x_(n)} - a*cos^2 x_(n)

i. Prove that there exist an open interval (a,b) c (o,pi/2) so if x_(0) is in (a,b) then the sequence is convergent.

ii. For x_(0) is in (a,b) find lim(x-->inf) {x_(n)}