If $x$ and $y$ are acute angles such that $\cos x + \cos y = \frac{3}{2}$ and $\sin x + \sin y = \frac{3}{4}$,then $\sin(x + y)$ equals to

If $m \cdot \tan (\theta-30^{\circ})=n \cdot \tan (\theta+120^{\circ})$,then $\frac{m+n}{m-n}=$

$\frac{\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 89^{\circ}}{2(\cos 1^{\circ}+\cos 2^{\circ}+\ldots+\cos 44^{\circ})+1} = $

If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ where $\alpha, \beta \in (0, \frac{\pi}{2})$,then $\tan (\alpha+2 \beta)$ is equal to

If $\tanh ^2 x = \tan ^2 \theta$,then $\cosh 2x =$

The value of $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ} = $