LHS=(sec8alpha-1)/(sec4alpha-1)
=(1/(cos8alpha)-1)/(1/(cos4alpha)-1)
=((1-cos8alpha)/(cos8alpha))/((1-cos4alpha)/(cos4alpha))
=(2sin^2 4alphacos4alpha)/(2sin^2 2 alphacos8alpha)
=(sin 8alpha*sin4alpha)/(2sin^2 2 alphacos8alpha)
=(sin 8alpha*2sin2alphacos2alpha)/(2sin^2 2 alphacos8alpha)
=(tan 8alpha)/((sin 2 alpha)/(cos 2alpha)
=(tan 8alpha)/(tan 2 alpha)=RHS