frac{sec8theta -1}{sec 4 theta -1} = frac{Tan 8 theta}{Tan 2 theta}

( begin{aligned} & frac{sec theta theta-1}{sec 4 theta-1}=left(frac{1-cos theta theta}{1-cos 4 theta}right) frac{cos 4 theta}{cos 80} &=frac{2 sin ^{2} 4 theta}{2 sin ^{2} 2 theta} cdot frac{cos 4 theta}{cos theta theta} &=frac{(2 sin 4 theta cos 4 theta) sin 40}{2 sin ^{2} 2 theta cos 8 theta}=frac{sin 80 sin 40}{2 sin ^{2} 2 theta cos theta theta} &=frac{2 tan theta theta sin 2 theta cos 2 theta}{2 sin ^{2} 2 theta}=tan 8 theta cos 2 theta &=frac{tan 8 theta}{tan 2 theta}=operatorname{RHS}  end{aligned} )