Suppose $x, y \in (0, \pi/2)$ and $x \neq y$. Which of the following statement is true?
(A) $2\sin(x + y) < \sin 2x + \sin 2y$ for all $x, y$.
(B) $2\sin(x + y) > \sin 2x + \sin 2y$ for all $x, y$.
(C) There exist $x, y$ such that $2\sin(x + y) = \sin 2x + \sin 2y$.
(D) None of the above.

Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true?
(a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$.
(b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$.
(c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$.
(d) None of the above.

Given $45^{\circ} < \theta < 50^{\circ}$, and let $a = 1 - \cos^{2}\theta$, $b = \frac{1}{\cos\theta} - \cos\theta$, $c = \frac{\tan\theta}{\tan^{2}\theta + 1}$. Regarding the relative sizes of the three values $a$, $b$, $c$, select the correct option.
(1) $a < b < c$
(2) $a < c < b$
(3) $b < a < c$
(4) $b < c < a$
(5) $c < a < b$

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) $-, -, +$