Let L denotes the value of the expression $\frac{\sin 2\theta -\sin 6\theta +\cos 2\theta -\cos 4\theta }{\sin 4\theta -\cos 4\theta }$,  when $\theta =27°$ and M denotes the value of $\frac{\tan x\tan 2x}{\tan 2x-\tan x}$ , when $x=9°$ and N denotes the numerical value of the expression $\frac{1-\cos 4\alpha }{{\sec }^{2}2\alpha -1}+\frac{1+\cos 4\alpha }{{\text{cosec}}^{2}2\alpha -1}$ then the value of the product $\text{LMN}=$

$\begin{array}{l}\text{L}=\frac{-2\cos 4\theta \sin 2\theta +2\sin 4\theta \sin 2\theta }{\sin 4\theta -\cos 4\theta }\\ =\frac{2\sin 2\theta (\sin 4\theta -\cos 4\theta )}{\sin 4\theta -\cos 4\theta }\\ =2\sin 2\theta \\ If\theta =27°,L=2\sin 54°=\frac{\sqrt{5}+1}{2}\\ \text{M}=\frac{\tan x\tan 2x}{\tan 2x-\tan x}\\ =\frac{\tan x\frac{2\tan x}{1-{\tan }^{2}x}}{\frac{2\tan x}{1-{\tan }^{2}x}-\tan x}\\ =\frac{2\tan x}{2-(1-{\tan }^{2}x)}\\ =\frac{2\tan x}{1+{\tan }^{2}x}\\ =\sin 2x\\ Whenx=9°,then\text{\hspace{0.17em}M}=\sin 18°=\frac{\sqrt{5}-1}{4}\\ \text{N}=\frac{1-\cos 4\alpha }{{\sec }^{2}2\alpha -1}+\frac{1+\cos 4\alpha }{{\text{cosec}}^{2}2\alpha -1}\\ =\frac{2{\sin }^{2}2\alpha {\cos }^{2}2\alpha }{1-{\cos }^{2}2\alpha }+\frac{2{\cos }^{2}2\alpha {\sin }^{2}2\alpha }{1-{\sin }^{2}2\alpha }\\ =2({\cos }^{2}2\alpha +{\sin }^{2}2\alpha )=2\\ \text{LMN}=\frac{\sqrt{5}+1}{2}\times \frac{\sqrt{5}-1}{4}\times 2=1\end{array}$