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 $\text{LMN}=$  

$=\frac{2\sin 2\theta \left(\sin 4\theta -\cos 4\theta \right)}{\sin 4\theta -\cos 4\theta }$

$=2\sin 2\theta$

If $\theta =27°,L=2\sin 54°=\frac{\sqrt{5}+1}{2}$

$\text{M}=\frac{\tan x\tan 2x}{\tan 2x-\tan x}$

$=\frac{\tan x\frac{2\tan x}{1-{\tan }^{2}x}}{\frac{2\tan x}{1-{\tan }^{2}x}-\tan x}$

$=\frac{2\tan x}{2-\left(1-{\tan }^{2}x\right)}$

$=\frac{2\tan x}{1+{\tan }^{2}x}$

$=\sin 2x$

When $x=9°$ ,  then $\text{M}=\sin 18°=\frac{\sqrt{5}-1}{4}$

$\text{N}=\frac{1-\cos 4\alpha }{{\sec }^{2}2\alpha -1}+\frac{1+\cos 4\alpha }{{\text{cosec}}^{2}2\alpha -1}$

$=\frac{2{\sin }^{2}2\alpha {\cos }^{2}2\alpha }{1-{\cos }^{2}2\alpha }+\frac{2{\cos }^{2}2\alpha {\sin }^{2}2\alpha }{1-{\sin }^{2}2\alpha }$

$=2\left({\cos }^{2}2\alpha +{\sin }^{2}2\alpha \right)=2$

$\text{LMN}=\frac{\sqrt{5}+1}{2}\times \frac{\sqrt{5}-1}{4}\times 2=1$