If $\sin A + \sin B = x$ and $\cos A + \cos B = y$,then $\sin(A + B) = $

The value of the series $\cos 12^{\circ} + \cos 84^{\circ} + \cos 132^{\circ} + \cos 156^{\circ}$ is

If $\sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta$ and $\alpha, \beta \in [0, \pi],$ then $\cos(\alpha + \beta)$ is equal to

If $a = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ and $x$ is the solution of the equations $y = 2[x] + 2$ and $y = 3[x - 2]$,where $[x]$ denotes the greatest integer function of $x$,then $a$ is equal to:

The value of $\tan 40^{\circ} + \tan 11^{\circ} + \tan 20^{\circ} - \tan 56^{\circ} + \tan 56^{\circ} \tan 11^{\circ} + \sqrt{3} \tan 40^{\circ} \tan 20^{\circ}$ is

If $\tan \theta + \cot \theta = 2$,then $\sin \theta$ is equal to