Given $\sin \alpha + \cos \beta = 1 , \cos \alpha + \sin \beta = 0$, then $\sin ( \alpha + \beta ) = \_\_\_\_$.

Given the function $f ( x ) = 2 \sin x + \sin 2 x$, the minimum value of $f ( x )$ is $\_\_\_\_$

10. Given $\alpha \in \left( 0 , \frac { \pi } { 2 } \right) , 2 \sin 2 \alpha = \cos 2 \alpha + 1$, then $\sin \alpha =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { 2 \sqrt { 5 } } { 5 }$

11. Given $a \in \left( 0 , \frac { \pi } { 2 } \right) , 2 \sin 2 \alpha = \cos 2 \alpha + 1$, then $\sin \alpha =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { 2 \sqrt { 5 } } { 5 }$

14. Given $\tan \left( \alpha + \frac { \pi } { 4 } \right) = 6$, then $\tan \alpha = $ \_\_\_\_.

If $\alpha$ is an angle in the fourth quadrant, then
A． $\cos 2 \alpha > 0$
B． $\cos 2 \alpha \leqslant 0$
C． $\sin 2 \alpha > 0$
D． $\sin 2 \alpha < 0$

Given $2 \tan \theta - \tan \left( \theta + \frac { \pi } { 4 } \right) = 7$ , then $\tan \theta =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

9. If $a \in \left(0, \frac{\pi}{2}\right)$, $\tan 2a = \frac{\cos a}{2 - \sin a}$, then $\tan a =$
A. $\frac{\sqrt{15}}{15}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{\sqrt{5}}{3}$
D. $\frac{\sqrt{15}}{3}$

11. If $\alpha \in \left( 0 , \frac { \pi } { 2 } \right) , \tan 2 \alpha = \frac { \cos \alpha } { 2 - \sin \alpha }$, then $\tan \alpha =$
A. $\frac { \sqrt { 15 } } { 15 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 5 } } { 3 }$
D. $\frac { \sqrt { 15 } } { 3 }$

Given that $\alpha$ is an acute angle and $\cos\alpha=\frac{1+\sqrt{5}}{4}$, then $\sin\frac{\alpha}{2}=$
A. $\frac{3-\sqrt{5}}{8}$
B. $\frac{-1+\sqrt{5}}{8}$
C. $\frac{3-\sqrt{5}}{4}$
D. $\frac{-1+\sqrt{5}}{4}$

Given $\cos ( \alpha + \beta ) = m , \tan \alpha \tan \beta = 2$ , then $\cos ( \alpha - \beta ) =$
A. $- 3 m$
B. $- \frac { m } { 3 }$
C. $\frac { m } { 3 }$
D. $3 m$

Given that $\alpha$ is an angle in the first quadrant, $\beta$ is an angle in the third quadrant, $\tan \alpha + \tan \beta = 4$, $\tan \alpha \tan \beta = \sqrt { 2 } + 1$, then $\sin ( \alpha + \beta ) =$ $\_\_\_\_$ .

Given $0 < a < \pi$, $\cos \frac{a}{2} = \frac{\sqrt{5}}{5}$, then $\sin\left(a - \frac{\pi}{4}\right) = $
A. $\frac{\sqrt{2}}{10}$
B. $\frac{\sqrt{2}}{5}$
C. $\frac{3\sqrt{2}}{10}$
D. $\frac{7\sqrt{2}}{10}$

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.
b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.
c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.
d) Specify the parity properties of $F_n$ as a function of $n$.

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Calculate $F_{n+1}(x) + F_{n-1}(x)$ for all $n \in \mathbb{N}^*$ and all $x \in D$.

Let $n \in \mathbb{N}$. Explicitly give a polynomial $P_n \in \mathbb{R}[X]$ such that, for all $\theta \in \mathbb{R}$, $$\sin((2n+1)\theta) = \sin(\theta) P_n\left(\sin^2(\theta)\right).$$ Hint: you may expand $(\cos(\theta) + i\sin(\theta))^{2n+1}$.

Let $n \in \mathbb{N}$ and let $P_n \in \mathbb{R}[X]$ be such that, for all $\theta \in \mathbb{R}$, $\sin((2n+1)\theta) = \sin(\theta) P_n\left(\sin^2(\theta)\right)$, and $$P_n(x) = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{x}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$
Deduce that, for all $x \in \mathbb{R}$, $$\sin(\pi x) = (2n+1)\sin\left(\frac{\pi x}{2n+1}\right) \prod_{k=1}^{n}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is the unique polynomial with real coefficients satisfying the relation $$\forall \theta \in \mathbb { R } , \quad T _ { n } ( \cos ( \theta ) ) = \cos ( n \theta ).$$

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Deduce that for all $x \in \mathbf{R}$, $$\Psi(x) = 2\ln\left(\frac{a+b}{2}\right) - 2\sum_{k=1}^{+\infty} \frac{\cos(2kx)}{k}\rho^k$$

We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Show that, for all $p \in \mathbb{N}^*$ and $x \in \mathbb{R} \backslash \pi\mathbb{Z}$, $$\sum_{k=1}^{p} \cos(2kx) = \frac{1}{2}\left(\frac{\sin((2p+1)x)}{\sin(x)} - 1\right)$$

We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Show that $$\sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) = \begin{cases} 0 & \text{if } p \text{ is even} \\ \dfrac { ( - 1 ) ^ { q + 1 } } { 2 } & \text{if } p \text{ is odd} \end{cases}$$

We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$. We admit that for all $0 \leq k \leq \lfloor p / 2 \rfloor - 1$, $$\int _ { 0 } ^ { 1 } F _ { k } ( t ) d t = \cos \left( q \theta _ { k } \right) \ln \left( 2 \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) - \frac { \pi } { 2 p } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right)$$
Deduce from the previous questions that, for all $( p , q ) \in E _ { 3 }$, $$S _ { p , q } = \frac { 1 } { p } \left( \frac { \pi } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right) - 2 \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) \ln \left( \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) \right)$$

Using the formula established for $( p , q ) \in E _ { 3 }$: $$S _ { p , q } = \frac { 1 } { p } \left( \frac { \pi } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } ( p - 1 - 2 k ) \sin \left( q \theta _ { k } \right) - 2 \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \cos \left( q \theta _ { k } \right) \ln \left( \sin \left( \frac { \theta _ { k } } { 2 } \right) \right) \right)$$ where $\theta_k := (2k+1)\dfrac{\pi}{p}$, deduce the exact values of $S _ { 2,1 }$ and $S _ { 3,1 }$.

The expression $$\sum _ { k = 0 } ^ { 10 } 2 ^ { k } \tan \left( 2 ^ { k } \right)$$ equals
(A) $\cot 1 + 2 ^ { 11 } \cot \left( 2 ^ { 11 } \right)$.
(B) $\cot 1 - 2 ^ { 10 } \cot \left( 2 ^ { 10 } \right)$.
(C) $\cot 1 + 2 ^ { 10 } \cot \left( 2 ^ { 10 } \right)$.
(D) $\cot 1 - 2 ^ { 11 } \cot \left( 2 ^ { 11 } \right)$.

Suppose, for some $\theta \in \left[ 0 , \frac { \pi } { 2 } \right] , \frac { \cos 3 \theta } { \cos \theta } = \frac { 1 } { 3 }$. Then $( \cot 3 \theta ) \tan \theta$ equals
(A) $\frac { 1 } { 2 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 8 }$
(D) $\frac { 1 } { 7 }$