Given \(\displaystyle \sin{x}\sin{y} = \dfrac{1}{2}\left(\cos{A} - \cos{B}\right)\)

i) Find \(A\) and \(B\) in terms of \(x, y\).
ii) Hence prove that for any positive integer \(n\):

\[ \sin{x}+\sin{3x}+\sin{5x}+\dots+\sin\left(2n-1\right)x = \dfrac{\sin^{2}nx}{\sin{x}}\]