Two right triangles $A B C$ and $B C D$ with one side coinciding are drawn as shown in the figure, and the resulting two regions are painted yellow and blue.
$$\mathrm { m } ( \widehat { \mathrm { DCA } } ) = \mathrm { m } ( \widehat { \mathrm { BAC } } ) = \mathrm { x }$$
Accordingly, what is the expression in terms of x for the ratio of the area of the yellow region to the area of the blue region?
A) $\sin 2 x$
B) $\cos 2 x$
C) $\sin ^ { 2 } x$
D) $\cot ^ { 2 } x$
E) $\csc ^ { 2 } x$

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

$$\frac { \cos ^ { 2 } \left( 80 ^ { \circ } \right) + 5 \sin ^ { 2 } \left( 80 ^ { \circ } \right) - 3 } { \cos \left( 50 ^ { \circ } \right) }$$
Which of the following is this expression equal to?
A) $\cot \left( 50 ^ { \circ } \right)$
B) $\sec \left( 20 ^ { \circ } \right)$
C) $\sec \left( 40 ^ { \circ } \right)$
D) $\operatorname { cosec } \left( 20 ^ { \circ } \right)$
E) $\operatorname { cosec } \left( 40 ^ { \circ } \right)$

In the figure, using the points $\mathrm { P } ( 0,1 )$ and $\mathrm { S } ( 1,0 )$ on the unit circle with center O and the positive directed angle $\theta$ that the line segment RO makes with the x-axis, new trigonometric functions are defined as follows:
$$\begin{aligned} & \text { kas } \theta = | \mathrm { RS } | \\ & \text { sas } \theta = | \mathrm { RP } | \end{aligned}$$
Accordingly,
$$\frac { \mathrm { kas } ^ { 2 } \theta } { 2 - \operatorname { sas } ^ { 2 } \theta }$$
For $\theta$ values where this expression is defined, which of the following is it equal to?
A) $\sin ( 2 \theta )$
B) $\cos ^ { 2 } ( 2\theta )$
C) $\sec \theta$
D) $\tan \left( \frac { \theta } { 2 } \right)$

$$\frac{1}{1 + \cot x} - \frac{\sin x}{\sin x - \cos x}$$
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 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) $\cot x$