Let $y=4{\sin }^2\theta -\cos 2\theta .$ If $l$ and $m$ are the minimum and maximum values of $y$ respectively, then

$y=4{\sin }^2\theta -\cos 2\theta =4{\sin }^2\theta -\left(1-2{\sin }^2\theta \right)$

$y=6{\sin }^2\theta -1$

$\therefore {\sin }^2\theta =\frac{y+1}{6}$

$\because 0\le {\sin }^2\theta \le 1$

$\Rightarrow 0\le \frac{y+1}{6}\le 1$

$\Rightarrow -1\le y\le 5$

Therefore, Minimum value of $y=l=-1$

And, Maximum value of $y=m=5$

Now, $lm=-5 \& \frac{m}{l}=-5$

Therefore, we can say that $lm=\frac{m}{l}$