The value of $\cos \,\frac{\pi }{7}\,\cos \,\frac{{2\pi }}{7}\,\cos \,\frac{{3\pi }}{7}$ is

For any $\theta \, \in \,\left( {\frac{\pi }{4},\frac{\pi }{2}} \right)$, the expression $3\,{\left( {\sin \,\theta  - \cos \,\theta } \right)^4} + 6{\left( {\sin \,\theta  + \cos \,\theta } \right)^2} + 4\,{\sin ^6}\,\theta $ equals

Prove that $\frac{\sin x-\sin 3 x}{\sin ^{2} x-\cos ^{2} x}=2 \sin x$

$\cos 20^\circ \cos 40^\circ \cos 80^\circ = $

The value of $\tan 7\frac{1}{2}^\circ $ is equal to

If $a{\sin ^2}x + b{\cos ^2}x = c,\,\,$$b\,{\sin ^2}y + a\,{\cos ^2}y = d$ and $a\,\tan x = b\,\tan y,$ then $\frac{{{a^2}}}{{{b^2}}}$ is equal to