The value of $ \cos ^{3}\left(\frac{\pi}{8}\right) \cdot \cos \left(\frac{3 \pi}{8}\right)+\sin ^{3}\left(\frac{\pi}{8}\right) \cdot \sin \left(\frac{3 \pi}{8}\right)$ is
$\frac{1}{4}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{2\sqrt{2}}$
$\frac{1}{2}$
Suppose $\theta $ and $\phi (\ne 0)$ 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
Let $S=\left\{x \in(-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set $S$ is equal to
$\frac{{\cos A}}{{1 - \sin A}} = $
${\sin ^4}\frac{\pi }{4} + {\sin ^4}\frac{{3\pi }}{8} + {\sin ^4}\frac{{5\pi }}{8} + {\sin ^4}\frac{{7\pi }}{8} = $
If $x = \sin {130^o}\,\cos {80^o},\,\,y = \sin \,{80^o}\,\cos \,{130^o},\,\,z = 1 + xy,$which one of the following is true