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Evaluate : $\frac{2}{3}(\cos^4 30^\circ - \sin^4 45^\circ) - 3(\sin^2 60^\circ - \sec^2 45^\circ) + \frac{1}{4}\cot^2 30^\circ$.
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$\frac{2}{3} [(\frac{\sqrt{3}}{2})^4 - (\frac{1}{\sqrt{2}})^4] - 3 [(\frac{\sqrt{3}}{2})^2 - (\sqrt{2})^2] + \frac{1}{4}(\sqrt{3})^2 = \frac{113}{24}$