When $\tan \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 2 }$, what is the value of $\sec \theta$? (where $0 < \theta < \frac { \pi } { 2 }$ ) [3 points]
(1) 3
(2) $\frac { 10 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) 4
(5) $\frac { 13 } { 3 }$

When $\tan \theta = 5$, find the value of $\sec ^ { 2 } \theta$. [3 points]

For $\theta$ satisfying $\pi < \theta < \frac { 3 } { 2 } \pi$ and $\tan \theta - \frac { 6 } { \tan \theta } = 1$, what is the value of $\sin \theta + \cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 10 } } { 5 }$
(2) $- \frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 10 } } { 5 }$
(5) $\frac { 2 \sqrt { 10 } } { 5 }$

If $\tan \theta < 0$ and $\cos \left( \frac { \pi } { 2 } + \theta \right) = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 5 } } { 5 }$
(2) $- \frac { \sqrt { 5 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 5 } } { 5 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$

When $\cos\left(\frac{\pi}{2} + \theta\right) = -\frac{1}{5}$, what is the value of $\frac{\sin\theta}{1 - \cos^{2}\theta}$? [3 points]
(1) $-5$
(2) $-\sqrt{5}$
(3) $0$
(4) $\sqrt{5}$
(5) $5$

When $\sin \theta + 3 \cos \theta = 0$ and $\cos ( \pi - \theta ) > 0$, what is the value of $\sin \theta$? [3 points]
(1) $\frac { 3 \sqrt { 10 } } { 10 }$
(2) $\frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $- \frac { \sqrt { 10 } } { 5 }$
(5) $- \frac { 3 \sqrt { 10 } } { 10 }$

4. The value of the determinant $\left| \begin{array} { c c } \cos \frac { \pi } { 3 } & \sin \frac { \pi } { 6 } \\ \sin \frac { \pi } { 3 } & \cos \frac { \pi } { 6 } \end{array} \right|$ is $\_\_\_\_$ $0$.
Analysis: This examines the rules for computing determinants. $\left| \begin{array} { c c } \cos \frac { \pi } { 3 } & \sin \frac { \pi } { 6 } \\ \sin \frac { \pi } { 3 } & \cos \frac { \pi } { 6 } \end{array} \right| = \cos \frac { \pi } { 3 } \cos \frac { \pi } { 6 } - \sin \frac { \pi } { 3 } \sin \frac { \pi } { 6 } = \cos \frac { \pi } { 2 } = 0$

Given $\alpha \in ( 0 , \pi )$ and $3 \cos 2 \alpha - 8 \cos \alpha = 5$, then $\sin \alpha =$
A. $\frac { \sqrt { 5 } } { 3 }$
B. $\frac { 2 } { 3 }$
C. $\frac { 1 } { 3 }$
D. $\frac { \sqrt { 5 } } { 9 }$

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

Let $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$g\left(\frac{x}{2}\right) + g\left(\frac{1+x}{2}\right) = 2g(x)$$

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.

Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the identity $\pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}$, show that: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the result $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$ deduce: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}\zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right)$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Show that for all $n \in \mathbb{N}$: $$\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{if } n \geqslant 1 \end{cases}$$

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then $$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)!\zeta(2n)}{2^{2n-1}\pi^{2n}}$$ Using the relation $\sum_{k=0}^{n} \frac{b_k}{k!(n+1-k)!} = 0$ for $n \geqslant 1$, calculate $b_2$, $b_4$ and $b_6$, then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.

110- What is $\cos\!\left(3\sin^{-1}\dfrac{2\sqrt{2}}{3}\right)$?
(1) $-\dfrac{23}{27}$ (2) $-\dfrac{19}{27}$ (3) $-\dfrac{5}{9}$ (4) $-\dfrac{4}{9}$

109- The value of $\dfrac{1}{\sin 15°} - \dfrac{1}{\cos 15°}$ is which of the following?
(1) $2$ (2) $\sqrt{6}$ (3) $2\sqrt{2}$ (4) $2\sqrt{3}$

110- The value of $\displaystyle\lim_{x \to \frac{3\pi}{4}} \dfrac{1-\tan^2 x}{\sqrt{1+\sin 2x}}$ is which of the following?
(1) $-2\sqrt{2}$ (2) $-\sqrt{2}$ (3) $\sqrt{2}$ (4) $2\sqrt{2}$

109. What is the value of $\tan\dfrac{11\pi}{4} + \sin\dfrac{15\pi}{4}\cos\dfrac{13\pi}{4}$?
(1) $-\dfrac{3}{2}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{3}{2}$

10. In the figure below, what is the value of $\cot\alpha$?
(1) $1$
(2) $2$
(3) $\dfrac{1}{2}$
(4) $\dfrac{1}{3}$
[Figure: A right triangle with a vertical side of length 2, and the base divided into two segments each of length 2, with angle $\alpha$ at the top left.]

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname{cosec} \frac{\pi}{n}$
(B) $r \left( 1 + \operatorname{cosec} \frac{2\pi}{n} \right)$
(C) $r \left( 1 + \operatorname{cosec} \frac{\pi}{2n} \right)$
(D) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$

If $\sin \left( \tan ^ { - 1 } ( x ) \right) = \cot \left( \sin ^ { - 1 } \left( \sqrt { \frac { 13 } { 17 } } \right) \right)$ then $x$ is
(A) $\frac { 4 } { 17 }$
(B) $\frac { 2 } { 3 }$
(C) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 ^ { 2 } + 13 ^ { 2 } } }$
(D) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 \times 13 } }$.

Find the minimum value of $$|\sin x + \cos x + \tan x + \cot x + \sec x + \operatorname{cosec} x|$$ for real numbers $x$ not multiple of $\pi/2$.