QUESTION 180
The value of $\cos^2 30^\circ + \sin^2 30^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sqrt{3}$
(E) 2

16. Given the function $f ( x ) = ( \sin x + \cos x ) ^ { 2 } + \cos 2 x$
(1) Find the minimum positive period of $f ( x )$;
(2) Find the maximum and minimum values of $f ( x )$ on the interval $\left[ 0 , \frac { \pi } { 2 } \right]$.

Given $\sin \theta + \sin \left( \theta + \frac { \pi } { 3 } \right) = 1$, then $\sin \left( \theta + \frac { \pi } { 6 } \right) =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 3 } } { 3 }$
C. $\frac { 2 } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$

Let $f(x) = \pi \operatorname{cotan}(\pi x)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x)$$

The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$ can be written as:
(1) $\tan A + \cot A$
(2) $\sec A + \operatorname{cosec} A$
(3) $\sin A \cos A + 1$
(4) $\sec A \operatorname{cosec} A + 1$

The value of $\cos ^ { 2 } 10 ^ { \circ } - \cos 10 ^ { \circ } \cos 50 ^ { \circ } + \cos ^ { 2 } 50 ^ { \circ }$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 4 } + \cos 20 ^ { \circ }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 3 } { 2 } \left( 1 + \cos 20 ^ { \circ } \right)$

The value of $\sin 10 ^ { \circ } \sin 30 ^ { \circ } \sin 50 ^ { \circ } \sin 70 ^ { \circ }$ is:
(1) $\frac { 1 } { 36 }$
(2) $\frac { 1 } { 16 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 32 }$

If $\sin ^ { 2 } \left( 10 ^ { \circ } \right) \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 50 ^ { \circ } \right) \sin \left( 70 ^ { \circ } \right) = \alpha - \frac { 1 } { 16 } \sin \left( 10 ^ { \circ } \right)$, then $16 + \alpha ^ { - 1 }$ is equal to $\_\_\_\_$.

Let $f ( \theta ) = 3 \left( \sin ^ { 4 } \left( \frac { 3 \pi } { 2 } - \theta \right) + \sin ^ { 4 } ( 3 \pi + \theta ) \right) - 2 \left( 1 - \sin ^ { 2 } 2 \theta \right)$ and $S = \left\{ \theta \in [ 0 , \pi ] : f ^ { \prime } ( \theta ) = - \frac { \sqrt { 3 } } { 2 } \right\}$. If $4 \beta = \sum _ { \theta \in S } \theta$ then $f ( \beta )$ is equal to
(1) $\frac { 11 } { 8 }$
(2) $\frac { 5 } { 4 }$
(3) $\frac { 9 } { 8 }$
(4) $\frac { 3 } { 2 }$

$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$ is equal to
(1) 3
(2) 1
(3) 4
(4) 2

The value of $\left( \sin 70 ^ { \circ } \right) \left( \cot 10 ^ { \circ } \cot 70 ^ { \circ } - 1 \right)$ is
(1) $2/3$
(2) 1
(3) 0
(4) $3/2$

$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$
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) $\arccos x$

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
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) $\csc ^ { 2 } x$

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

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