If $A$ is in the third quadrant and $\tan A = \frac{\sqrt{7}}{3}$,then $18 - 16 \sin^2 \frac{A}{2} - 32 \sin \frac{A}{2} \sin \frac{5A}{2} = $

The value of $\frac{1}{4} \tan \frac{\pi}{8} + \frac{1}{8} \tan \frac{\pi}{16} + \frac{1}{16} \tan \frac{\pi}{32} + \dots \infty$ terms is equal to-

If $\cos A = m \cos B,$ then

If $\sin x + \cos x = a$,where $a \in [-\sqrt{2}, \sqrt{2}] - \{-1, 1\}$,then $\sum_{n=1}^{\infty} (\sin^n x + \cos^n x)$ is equal to -

Let $\alpha = \frac{1}{\sin 60^{\circ} \sin 61^{\circ}} + \frac{1}{\sin 62^{\circ} \sin 63^{\circ}} + \dots + \frac{1}{\sin 118^{\circ} \sin 119^{\circ}}$. Then the value of $\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2$ is $....$

$\frac{1-\cos(2x)+\sin(x)}{\sin(2x)+\cos(x)} = $