Çözüldü Trigonometry - Ratio of Rectangle's Area to Triangle's Area

For a semi-circle with the center O, diameter [AB], two tangents [AB] and [CD] making a rectangle ABCD, if [BC] intersects the circle at the points E and F, what is the ratio of the area of the ABCD rectangle to that of EFO with respect to the central angle ∡EOF = x ?

The radius of the Circle: r units
Area of the ABCD rectangle: 2r·{ r·sin[ 90° - (x / 2) ] } = 2r^2·cos(x / 2)....(I)
Area of the ΔEFO: (1 / 2)·r·r·sin(x) = (1 / 2)·r^2·2·sin(x / 2)·cos(x / 2) = r^2·sin(x / 2)·cos(x / 2)....(II)
Dividing (I) by (II) and simplifying 2 / sin(x / 2) = 2·csc(x / 2).

Note: Many thanks to our new member, who gave the following pictorial solution.