$tan\, 20^{\circ} + tan\, 40^{\circ} + \sqrt{3}\, tan\, 20^{\circ} tan\, 40^{\circ}$ is equal to

Two ships leave a port from a point at the same time. One goes with a velocity of $3 \text{ km/h}$ along North-East making an angle of $45^{\circ}$ with the East direction and the other travels with a velocity of $4 \text{ km/h}$ along South-East making an angle of $15^{\circ}$ with the East direction. Then,the distance between the ships at the end of two hours is

If $\cos \alpha + \cos \beta = \frac{1}{3}$ and $\sin \alpha + \sin \beta = \frac{1}{4}$,then $\cos (\alpha + \beta) = $

If $A=35^{\circ}, B=15^{\circ}$ and $C=40^{\circ}$,then $\tan A \cdot \tan B+\tan B \cdot \tan C+\tan C \cdot \tan A$ is equal to

If $\sin x = \frac{3}{5}$ and $\cos y = -\frac{12}{13}$,where $x$ and $y$ both lie in the second quadrant,find the value of $\sin (x+y)$.

$\cos ^2 48^{\circ}-\sin ^2 12^{\circ} = $ . . . . . . ,if $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$