$\cos \frac{\pi }{5}\cos \frac{{2\pi }}{5}\cos \frac{{4\pi }}{5}\cos \frac{{8\pi }}{5} = $

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then $\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)$ is equal to

If $A$ lies in the third quadrant and $3\,\tan A - 4 = 0,$ then $5\,\sin 2A + 3\,\sin A + 4\,\cos A = $

If $x\cos \theta = y\cos \,\left( {\theta + \frac{{2\pi }}{3}} \right) = z\cos \,\left( {\theta + \frac{{4\pi }}{3}} \right),$ then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

If $a\,\cos 2\theta + b\,\sin 2\theta = c$  has $\alpha$ and $\beta$ as its solution, then the value of $\tan \alpha + \tan \beta $ is

If $\sin \alpha = \frac{{336}}{{625}}$ and $450^\circ < \alpha < 540^\circ ,$ then $\sin \left( {\frac{\alpha }{4}} \right) = $