If Z = cos θ + i sin θ, (θ belongs to set of real numbers, i = √-1), then according to DeMoivre's theorem, Z^n = cos nθ + i sin nθ, (n belongs to set of natural numbers). Using this theorem alongwith binomial expansion, prove that:

(i) cos 2nθ =∑( r = 0 to n) (2n2r) cos^(2(n - r)) θ sin^(2r) θC

(ii) cos 2nθ =∑(r = 0 to n) (2n(2r + 1))cos^(2(n-r) - 1) θ sin^(2r + 1) θC

NOTE: 2n2r is 2n choose 2r.C