$tan2x+cotx=8cos^2x$$\Leftrightarrow \frac{sin2x}{cos2x}+\frac{cosx}{sinx}-8cos^2x=0$
$\Leftrightarrow cosx(\frac{2sinx}{cos2x}+\frac{1}{sinx}-8cosx)=0$
$\Leftrightarrow cosx(\frac{2sin^2x+cos2x-8sinx.cosx.cos2x}{cos2x.sinx})=0$
$\Leftrightarrow cosx(\frac{1-2sin4x}{cos2x.sinx})=0$