The value of $2 \sin(\frac{\pi}{8}) \sin (\frac{2 \pi}{8}) \sin (\frac{3 \pi}{8}) \sin (\frac{5 \pi}{8}) \sin (\frac{6 \pi}{8}) \sin (\frac{7 \pi}{8})$ is:

The value of $\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$ is $............$.

If ${\cos ^6}\alpha + {\sin ^6}\alpha + K\,{\sin ^2}2\alpha = 1,$ then $K =$

If $\tan \theta = \frac{{\sin \alpha - \cos \alpha }}{{\sin \alpha + \cos \alpha }},$ then $\sin \alpha + \cos \alpha $ and $\sin \alpha - \cos \alpha $ must be equal to

${\cos ^2}A{(3 - 4{\cos ^2}A)^2} + {\sin ^2}A{(3 - 4{\sin ^2}A)^2} = $

$\left( {1 + \cos \frac{\pi }{8}} \right)\,\left( {1 + \cos \frac{{3\pi }}{8}} \right)\,\left( {1 + \cos \frac{{5\pi }}{8}} \right)\,\left( {1 + \cos \frac{{7\pi }}{8}} \right) = $