If $\cos A = \frac{3}{4}$,then $32\sin \frac{A}{2}\cos \frac{5A}{2} = $

If $\tan A = \frac{1 - \cos B}{\sin B}$,find $\tan 2A$ in terms of $\tan B$ and show that:

If $\sin \theta + \sin^2 \theta = 1$,then $\cos^2 \theta + \cos^4 \theta$ 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 $\text{m}$):

$\cos^2 \alpha + \cos^2(\alpha + 120^\circ) + \cos^2(\alpha - 120^\circ)$ is equal to

If $\cos \alpha = \frac{2 \cos \beta - 1}{2 - \cos \beta}$,then the value of $\tan \frac{\alpha}{2} \cot \frac{\beta}{2}$ is,where $(0 < \alpha < \pi$ and $0 < \beta < \pi$).