Answer the following questions
(a) Evaluate $$\lim_{x \rightarrow 0^{+}} \left( x^{x^{x}} - x^{x} \right)$$ (b) Let $A = \frac{2\pi}{9}$, i.e., $A = 40$ degrees. Calculate the following $$1 + \cos A + \cos 2A + \cos 4A + \cos 5A + \cos 7A + \cos 8A$$ (c) Find the number of solutions to $e^{x} = \frac{x}{2017} + 1$.

9. If $\tan \alpha = 2 \tan \frac { \pi } { 5 }$, then $\frac { \cos \left( \alpha - \frac { 3 \pi } { 10 } \right) } { \sin \left( \alpha - \frac { \pi } { 5 } \right) } =$
A. $1$
B. $2$
C. $3$
D. $4$

12. $\sin 15 ^ { \circ } + \sin 75 ^ { \circ } = \_\_\_\_$.

106- If $f(\alpha) = 4\sin(\alpha)\cos(2\alpha) + 2\sin(\alpha)$, what is the value of $f\!\left(\dfrac{41\pi}{9}\right)$?
(1) $-\sqrt{3}$ (2) $\sqrt{3}$ (3) $1$ (4) $-1$

The value of the limit $$\lim_{x \rightarrow \frac{\pi}{2}} \frac{4\sqrt{2}(\sin 3x + \sin x)}{\left(2\sin 2x \sin\frac{3x}{2} + \cos\frac{5x}{2}\right) - \left(\sqrt{2} + \sqrt{2}\cos 2x + \cos\frac{3x}{2}\right)}$$ is $\_\_\_\_$

The value of $\cos 255 ^ { \circ } + \sin 195 ^ { \circ }$ is
(1) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(2) $\frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(3) $- \frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }$

The value of $\cos \frac { \pi } { 2 ^ { 2 } } \cdot \cos \frac { \pi } { 2 ^ { 3 } } \cdot \ldots \cdot \cos \frac { \pi } { 2 ^ { 10 } } \cdot \sin \frac { \pi } { 2 ^ { 10 } }$ is:
(1) $\frac { 1 } { 1024 }$
(2) $\frac { 1 } { 512 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 256 }$

The value of $\cos ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \cos \left( \frac { 3 \pi } { 8 } \right) + \sin ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \sin \left( \frac { 3 \pi } { 8 } \right)$ is:
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$

The value of $\cot \frac { \pi } { 24 }$ is:
(1) $\sqrt { 2 } + \sqrt { 3 } + 2 - \sqrt { 6 }$
(2) $\sqrt { 2 } + \sqrt { 3 } + 2 + \sqrt { 6 }$
(3) $\sqrt { 2 } - \sqrt { 3 } - 2 + \sqrt { 6 }$
(4) $3 \sqrt { 2 } - \sqrt { 3 } - \sqrt { 6 }$

$16 \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 80 ^ { \circ } \right)$ is equal to
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) 3
(4) $4 \sqrt { 3 }$

The value of $2\sin 12^\circ - \sin 72^\circ$ is
(1) $\frac{\sqrt{5}(1 - \sqrt{3})}{4}$
(2) $\frac{1 - \sqrt{5}}{8}$
(3) $\frac{\sqrt{3}(1 - \sqrt{5})}{2}$
(4) $\frac{\sqrt{3}(1 - \sqrt{5})}{4}$

The value of $2\sin\frac{\pi}{22}\sin\frac{3\pi}{22}\sin\frac{5\pi}{22}\sin\frac{7\pi}{22}\sin\frac{9\pi}{22}$ is equal to:
(1) $\frac{1}{16}$
(2) $\frac{5}{16}$
(3) $\frac{7}{16}$
(4) $\frac{3}{16}$

The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $\_\_\_\_$.

The value of $36 \left( 4 \cos ^ { 2 } 9 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 27 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 81 ^ { \circ } - 1 \right) \left( 4 \cos ^ { 2 } 243 ^ { \circ } - 1 \right)$ is
(1) 54
(2) 18
(3) 27
(4) 36

If $\frac { \cos ^ { 2 } 48 ^ { \circ } - \sin ^ { 2 } 12 ^ { \circ } } { \sin ^ { 2 } 24 ^ { \circ } - \sin ^ { 2 } 6 ^ { \circ } } = \frac { \alpha + \sqrt { 5 } \beta } { 2 }$. Then, the value of $( \alpha + \beta )$ is
(A) 3
(B) 2
(C) 11
(D) 4

4. Which of the following numbers is closest to $\sqrt { 2 }$?
(1) $\sqrt { 3 } \cos 44 ^ { \circ } + \sin 44 ^ { \circ }$
(2) $\sqrt { 3 } \cos 54 ^ { \circ } + \sin 54 ^ { \circ }$
(3) $\sqrt { 3 } \cos 64 ^ { \circ } + \sin 64 ^ { \circ }$
(4) $\sqrt { 3 } \cos 74 ^ { \circ } + \sin 74 ^ { \circ }$
(5) $\sqrt { 3 } \cos 84 ^ { \circ } + \sin 84 ^ { \circ }$

$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$
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