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

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

\(\frac{sec²60º cos²45º + cosec²30º}{cot30º.sec²45º - cosec²30º.tan45º}\)

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

= \(\frac{6 }{2√3  - 4}\)

= \(\frac{3 }{√3  - 2}\)

Divide and multiply by ( √3 + 2 ).

= \(\frac{3 }{√3  - 2}\) × \(\frac{√3+ 2 }{√3 + 2}\)

= \(\frac{3 ( √3+ 2 )  }{3 - 4}\)

= - 3 ( √3+ 2 )