If $x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)$,then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

If $\sin A = \sin B$ and $\cos A = \cos B,$ then

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

If $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} = \frac{\sin \frac{8 \pi}{7}}{8 \sin \frac{\pi}{7}}$,then $\sin \frac{\pi}{14} \sin \frac{3 \pi}{14} \sin \frac{5 \pi}{14} \sin \frac{7 \pi}{14} \sin \frac{9 \pi}{14} \sin \frac{11 \pi}{14} \sin \frac{13 \pi}{14} = $

If $\cot \theta + \tan \theta = m$ and $\sec \theta - \cos \theta = n$,then which of the following is correct?

The geometric mean of $\tan 1^{\circ}, \tan 2^{\circ}, \ldots, \tan 89^{\circ}$ is