The value of $\cot(7.5^{\circ}) + \tan(67.5^{\circ}) - \cot(67.5^{\circ}) - \tan(7.5^{\circ})$ is:

If $\sin \theta = -\frac{7}{25}$ and $\theta$ is in the third quadrant,then $\frac{7 \cot \theta - 24 \tan \theta}{7 \cot \theta + 24 \tan \theta} = $

If $x = a \operatorname{cosec}^{n} \theta$ and $y = b \cot^{n} \theta$,then by eliminating $\theta$,we get:

${\left( {\frac{{\cos A + \cos B}}{{\sin A - \sin B}}} \right)^n} + {\left( {\frac{{\sin A + \sin B}}{{\cos A - \cos B}}} \right)^n}$ ($n$ is even or odd) $=$

The value of $\cos 41^{\circ} \cdot \cos 42^{\circ} \cdot \cos 43^{\circ} \cdot \cos 44^{\circ} \cdot \cos 45^{\circ} \cdot \cos 46^{\circ} \cdot \cos 47^{\circ} \cdot \cos 48^{\circ} \cdot \cos 49^{\circ}$ is:

In a $\Delta ABC$,$\angle B = \frac{\pi}{3}$,$\angle C = \frac{\pi}{4}$ and $D$ divides $BC$ internally in the ratio $1:3$,then $\frac{\sin \angle BAD}{\sin \angle CAD}$ is equal to