If  $\tan \frac{\pi }{9},x,\tan \frac{5\pi }{18}$ are in A.P and $\cot \frac{\pi }{9},y,\cot \frac{7\pi }{18}$ are in H.P then $8xy=$  ____

Given $\tan \frac{\pi }{9}, x, \tan \frac{5\pi }{18}$ are in A.P and $\cot \frac{\pi }{9}.y.\cot \frac{7\pi }{18}$  are in H.P then
$\tan 20+\tan 50=2x,\tan 20+\tan 70=\frac{2}{y}$
$\Rightarrow 2\tan 50=\tan 70-\tan 20$ adding $2\tan 20$ both sides
 $4x=\frac{2}{y}\Rightarrow xy=\frac{1}{2}$