$$\frac{1}{1 + \cot x} - \frac{\sin x}{\sin x - \cos x}$$
What is the simplified form of this expression?
A) $\sec(2x)$ B) $\sec^{2}(2x)$ C) $\tan(2x)$ D) $2 \cdot \sec x$ E) $2 \cdot \tan x$

In Figure 1, a parallelogram with two sides of lengths 4 units and 8 units and the angle between these sides measuring $x$ degrees is given. In Figure 2, a parallelogram with two sides of lengths 4 units and 6 units and the angle between these sides measuring $2x$ degrees is given.
If the area of the parallelogram in Figure 1 is 24 square units, what is the area of the parallelogram in Figure 2 in square units?
A) $6\sqrt{7}$ B) $7\sqrt{7}$ C) $8\sqrt{7}$ D) $9\sqrt{7}$ E) $10\sqrt{7}$

In the figure, a semicircle with center $O$ and diameter $AD$, a rectangle $ABCD$, and a triangle $OEF$ are given. Points $C$, $F$, $E$, $B$ are collinear; points $E$ and $F$ are on the circle.
Accordingly, what is the ratio of the area of rectangle $ABCD$ to the area of triangle $OEF$ in terms of $x$?
A) $\tan\frac{x}{2}$ B) $2 \cdot \sec x$ C) $2 \cdot \operatorname{cosec}\frac{x}{2}$ D) $2 \cdot \tan x$ E) $\cot x$