If $A$ and $B$ are positive acute angles satisfying $3 \cos^2 A + 2 \cos^2 B = 4$ and $\frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}$,then $A + 2B =$ (in $^{\circ}$)

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4} < \beta < 0 < \alpha < \frac{\pi}{4}$. If $\sin (\alpha+\beta) = \frac{1}{3}$ and $\cos (\alpha-\beta) = \frac{2}{3}$,then the greatest integer less than or equal to $\left(\frac{\sin \alpha}{\cos \beta} + \frac{\cos \beta}{\sin \alpha} + \frac{\cos \alpha}{\sin \beta} + \frac{\sin \beta}{\cos \alpha}\right)^2$ is:

$\sin 12^\circ \sin 24^\circ \sin 48^\circ \sin 84^\circ = $

Prove that: $(\sin 3x + \sin x) \sin x + (\cos 3x - \cos x) \cos x = 0$

The value of $2 \sin(12^{\circ}) - \sin(72^{\circ})$ is

If $x = 3 \sin \theta$,$y = 3 \cos \theta \cos \phi$,and $z = 3 \cos \theta \sin \phi$,then $x^{2} + y^{2} + z^{2} =$