If $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right)$,then:

$\frac{\sin 81^\circ + \cos 81^\circ}{\sin 81^\circ - \cos 81^\circ}$ is equal to

$A$ tree breaks at a certain height and the upper part touches the ground,making an angle of $60^{\circ}$ with the ground at a distance of $10 \text{ m}$ from its foot. The original height of the tree was (in $\text{m}$):

If $\alpha$ is a root of $25\cos^2\theta + 5\cos\theta - 12 = 0$ and $\pi/2 < \alpha < \pi$,then $\sin 2\alpha$ is equal to

Suppose $\theta$ and $\phi (\neq 0)$ are such that $\sec(\theta + \phi)$,$\sec\theta$,and $\sec(\theta - \phi)$ are in $A.P.$ If $\cos\theta = k \cos(\frac{\phi}{2})$ for some $k$,then $k$ is equal to

If $5 \sin \theta = 3$,find the numerical value of $\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta}$.