$$3\sin x - 4\cos x = 0$$
Given this, what is the value of $|\cos 2x|$?
A) $\frac{3}{4}$
B) $\frac{3}{5}$
C) $\frac{4}{5}$
D) $\frac{7}{25}$
E) $\frac{9}{25}$

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
Which of the following is this expression equal to?
A) $\frac{1}{\cos x}$
B) $\frac{1}{\sin x}$
C) $1$
D) $\arcsin x$
E) $\arccos x$

$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$
Which of the following is this expression equal to?
A) 4
B) 2
C) 1
D) $\frac{\sqrt{3}}{2}$
E) $\frac{1}{2}$

$$\frac{1+\cos 40^{\circ}}{\cos 55^{\circ} \cdot \cos 35^{\circ}}$$
Which of the following is this expression equal to?
A) $\cos 20^{\circ}$
B) $2\cos 20^{\circ}$
C) $4\cos 20^{\circ}$
D) $\cos 40^{\circ}$
E) $2\cos 40^{\circ}$

$$\frac { \sin 48 ^ { \circ } } { \sin 16 ^ { \circ } } - \frac { \cos 48 ^ { \circ } } { \cos 16 ^ { \circ } }$$
Which of the following is this expression equal to?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 4 } { 3 }$
D) 2
E) 3

ABCD is a square $| \mathrm { AB } | = 3$ units $| \mathrm { BE } | = | \mathrm { CF } | = 1$ unit $m ( \widehat { F A E } ) = x$
According to the given information above, what is the value of $\cot \mathrm { x }$?
A) $\frac { 6 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 7 } { 6 }$
D) $\frac { 9 } { 7 }$
E) $\frac { 11 } { 8 }$

In the regular pentagon ABCDE shown, K and L are the midpoints of line segments AB and DA, respectively.
Given this, what is the measure of angle LKB in degrees?
A) 105
B) 108
C) $\mathbf { 1 2 0 }$
D) 126
E) 135

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
What is the equivalent of this expression?
A) $2 \cot \left( 22 ^ { \circ } \right)$ B) $2 \cos \left( 56 ^ { \circ } \right)$ C) $4 \sin \left( 44 ^ { \circ } \right)$ D) $4 \cos \left( 34 ^ { \circ } \right)$ E) $4 \tan \left( 56 ^ { \circ } \right)$

Below are shown a semicircle with center O and radius 1 unit, and right triangles OAB and ODC. Points A and C lie on both the triangle OAB and the semicircle.
Accordingly, $$\frac { | \mathrm { AB } | + | \mathrm { BC } | } { | \mathrm { CD } | + | \mathrm { DA } | }$$
What is the equivalent of this ratio in terms of x?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\csc x$ E) $\sec x$

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) $\csc ^ { 2 } x$

In the figure, the line segments $[OA]$ and $[OD]$ intersect perpendicularly.
Accordingly, the ratio of the area of triangle OAB to the area of triangle OCD in terms of $\alpha$ is which of the following?
A) $\tan \alpha$
B) $\cot \alpha$
C) $\tan^2 \alpha$
D) $\cot^2 \alpha$
E) $\sec^2 \alpha$

$$\frac { \cos ^ { 2 } \left( 80 ^ { \circ } \right) + 5 \sin ^ { 2 } \left( 80 ^ { \circ } \right) - 3 } { \cos \left( 50 ^ { \circ } \right) }$$
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) $\tan \left( \frac { \theta } { 2 } \right)$

Let $a \in \left(\frac{3\pi}{4}, \pi\right)$,
$$\begin{aligned} & x = \sin(2a) \cdot \tan(a) \\ & y = \cos(2a) \cdot \cot(2a) \\ & z = \sin(a) \cdot \cot(2a) \end{aligned}$$
Given these equalities.
Accordingly, what are the signs of $\mathbf{x}$, $y$ and $\mathbf{z}$ respectively?
A) $+, +, -$ B) $+, -, -$ C) $-, -, -$ D) $-, +, +$ E) $-, -, +$

$$\frac{1}{1 + \cot x} - \frac{\sin x}{\sin x - \cos x}$$
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) $10\sqrt{7}$

Let $0 < a < \dfrac{\pi}{2}$ and
$$\cos^{2} a - \cos(2a) = \sin(2a)$$
For the value of $a$ satisfying this equality, which of the following is correct?
A) $\tan a = \dfrac{1}{5}$ B) $\cot a = \dfrac{2}{\sqrt{5}}$ C) $\cos a = \dfrac{1}{\sqrt{5}}$ D) $\operatorname{cosec} a = \sqrt{5}$ E) $\sin(2a) = \dfrac{3}{5}$