Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$

Which of the following number(s) is /are rational?

The number of values of $$x\,\,$$ in the interval $$\left[ {0,\,5\pi } \right]$$ satisfying the equation $$3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0$$ is

If $$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$$ , then