$$ \begin{aligned} & 4 \cos ^2 \theta-1=4\left(1-\sin ^2 \theta\right)-1 \\\\ & =3-4 \sin ^2 \theta \\\\ & =\frac{3 \sin \theta-4 \sin ^3 \theta}{\sin \theta} \\\\ & =\frac{\sin 3 \theta}{\sin \theta} \end{aligned} $$

$$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)$$

$$ \begin{aligned} & =36\left[\frac{\sin 27^{\circ}}{\sin 9^{\circ}} \times \frac{\sin 81^{\circ}}{\sin 27^{\circ}} \times \frac{\sin 243^{\circ}}{\sin 81^{\circ}} \times \frac{\sin 729^{\circ}}{\sin 243^{\circ}}\right] \\\\ & =36\left[\frac{\sin 729^{\circ}}{\sin 9^{\circ}}\right]=36 \times 1=36 \end{aligned} $$