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

If $x + y = 3 - cos4\theta$ and $x - y = 4 \,sin2\theta$ then

If $\cos \left( {\alpha + \beta } \right) = \frac{4}{5}$ and $\sin \left( {\alpha - \beta } \right) = \frac{5}{{13}}$,where $0 \le \alpha ,\beta \le \frac{\pi }{4}$ . Then $\tan 2\alpha =$ 

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

If $\cos A = \frac{3}{4}$, then $32\sin \frac{A}{2}\cos \frac{5}{2}A = $

$\frac{1}{{\sin 10^\circ }} - \frac{{\sqrt 3 }}{{\cos 10^\circ }} =$