The value of $\frac{4 \sin ^2 30^{\circ} \tan 60^{\circ}-3 \cos ^2 60^{\circ} \sec ^2 30^{\circ}}{4 \cot ^2 45^{\circ}-\sec ^2 60^{\circ}+\sin ^2 60^{\circ}+\cos ^2 90^{\circ}}$ is:

$\frac{2(\sqrt{3}-2)}{3}$

\frac{2 \sin ^2 30^{\circ} \tan 60^{\circ}-3 \cos ^2 60^{\circ} \sec ^2 30^{\circ}}{4 \cot ^2 45^{\circ}-\sec ^2 60^{\circ}+\sin ^2 60^{\circ}+\cos ^2 90^{\circ}}$

= \(\frac{ 2 × 1/4 × √3 - 3× 1/4 × 4/3}{ 4× 1 - 4 + 3/4 + 0 }\)

= \(\frac{ √3/2 - 1}{ 3/4 }\)

= \(\frac{2 (√3 - 2)}{ 3 }\)