hoanganhtuan96vn

Chứng minh rằng: $$\sqrt{1+\sqrt{1-x^{2}}}\left [ \sqrt{1+x^{3}}-\sqrt{1-x^{3}} \right ]\leq 2\sqrt{2}+\sqrt{2+2x^{2}}$$

1. $\left\{\begin{matrix} 2x^{3}+xy+y=5 & \\ x^{4}+x^{3}y+x^{2}(y+1)+xy+y=9 & \end{matrix}\right.$
2. $\left\{\begin{matrix} x+\frac{2}{y}=y+\frac{2}{x}& \\ \sqrt{x+8}-\sqrt{2y-2}=\sqrt{3y-2}& \end{matrix}\right.$
3. $\left\{\begin{matrix} (x-y)(x^{2}+xy+y^{2}+3)=3(x^{2}+y^{2})+2 & \\ 4\sqrt{x+2}+\sqrt{16-3y}=x^{2}+8& \end{matrix}\right.$

$1, tan^{2}A+tan^{2}B+tan^{2}C\leq cot^{2}\frac{A}{2}+cot^{2}\frac{B}{2}+cot^{2}\frac{C}{2}.$
$2, sin^{2}A+sin^{2}B+sin^{2}C=cos^{2}\frac{A}{2}+cos^{2}\frac{B}{2}+cos^{2}\frac{C}{2}$

$1, \frac{cos(B-C)}{sinA + cos(B-C))}=tanB$
$2, \frac{sinA+cosB}{sinB+cosA}=tanA$
$3, sin\frac{B}{2}=\sqrt{\frac{a-c}{a.c}}$
$4, cos(B-C)=\frac{2bc}{a^{2}}$

Cho x,y $\epsilon \begin{bmatrix} 0 ;\frac{\sqrt{2}}{2} \end{bmatrix}$:
Tìm Max S = $\frac{x}{1+y^{2}} + \frac{y}{1+x^{2}}$