$\cos \frac{\pi }{7}\cos \frac{{2\pi }}{7}\cos \frac{{4\pi }}{7} = $

If $a\tan \theta = b$, then $a\cos 2\theta + b\sin 2\theta = $

If $\cos A = \cos B\,\,\cos C$and $A + B + C = \pi ,$ then the value of $\cot \,B\,\cot \,C$ is

If ${\rm{cosec}}\theta = \frac{{p + q}}{{p - q}},$ then $\cot \,\left( {\frac{\pi }{4} + \frac{\theta }{2}} \right) = $

If $\cos \,(\theta - \alpha ) = a,\,\,\sin \,(\theta - \beta ) = b,\,\,$then ${\cos ^2}(\alpha - \beta ) + 2ab\,\sin \,(\alpha - \beta )$ is equal to

The value of ${\cos ^2}\,{10^o}\,\, - \,\cos \,\,{10^o}\,\cos \,\,{50^o}\, + \,{\cos ^2}\,{50^o}$ is