$$\frac{4}{3} \tan^2 60^\circ + 3 \cos^2 30^\circ - 2 \sec^2 30^\circ - \frac{3}{4} \cot^2 60^\circ$$ is equal to:

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

$$= \dfrac{4}{3}\times(\sqrt{3})^2 + 3 \times (\dfrac{\sqrt{3}}{2})^2 - 2\times(\dfrac{2}{\sqrt{3}})^2 - \dfrac{3}{4}\times(\dfrac{1}{\sqrt{3}})^2$$

$$= \dfrac{4}{3}\times3+3\times\dfrac{3}{4}-2\times\dfrac{4}{3}-\dfrac{3}{4}\times\dfrac{1}{3}$$

$$= 4+\dfrac{9}{4}-\dfrac{8}{3}-\dfrac{1}{4}$$

$$= \dfrac{10}{3}$$