The diagram shows a trapezium \(ABCD\), where \(AB\) is parallel to \(DC\), \(AD\) is perpendicular to \(DC\),

\(AB=2+3\sqrt5,\qquad DC=6+3\sqrt5,\qquad AD=10-2\sqrt5.\)

(i) Find the area of the trapezium in the form \(a+b\sqrt5\), where \(a\) and \(b\) are integers.

(ii) Find \(\operatorname{cot} BCD\) in the form \(c+d\sqrt5\), where \(c\) and \(d\) are constants.