Evaluate each trigonometric function.

We need to find the values of $\sin\left(\frac{2\pi}{3}\right)$, $\tan\left(\frac{5\pi}{3}\right)$, $\cot\left(\frac{-4\pi}{3}\right)$, and $\cos\left(\frac{5\pi}{6}\right)$.

For $\sin\left(\frac{2\pi}{3}\right)$:
The angle $\frac{2\pi}{3}$ is in the second quadrant.

$$\sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$

For $\tan\left(\frac{5\pi}{3}\right)$:
The angle $\frac{5\pi}{3}$ is in the fourth quadrant.

$$\tan\left(\frac{5\pi}{3}\right) = \tan\left(2\pi - \frac{\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}$$

For $\cot\left(\frac{-4\pi}{3}\right)$:
First, use the identity $\cot(-x) = -\cot(x)$:

$$\cot\left(\frac{-4\pi}{3}\right) = -\cot\left(\frac{4\pi}{3}\right)$$

The angle $\frac{4\pi}{3}$ is in the third quadrant.

$$\cot\left(\frac{4\pi}{3}\right) = \cot\left(\pi + \frac{\pi}{3}\right) = \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}}$$

So,

$$\cot\left(\frac{-4\pi}{3}\right) = -\frac{1}{\sqrt{3}}$$

For $\cos\left(\frac{5\pi}{6}\right)$:
The angle $\frac{5\pi}{6}$ is in the second quadrant.

$$\cos\left(\frac{5\pi}{6}\right) = \cos\left(\pi - \frac{\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$$

$$A = \sin\left(\frac{2\pi}{3}\right) \tan\left(\frac{5\pi}{3}\right) \cot\left(\frac{-4\pi}{3}\right) \cos\left(\frac{5\pi}{6}\right)$$ $$A = \left(\frac{\sqrt{3}}{2}\right) \cdot (-\sqrt{3}) \cdot \left(-\frac{1}{\sqrt{3}}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right)$$

$$A = \frac{\sqrt{3}}{2} \cdot \left(-\frac{\sqrt{3}}{1}\right) \cdot \left(-\frac{1}{\sqrt{3}}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right)$$

Multiply the numerators:

$$\sqrt{3} \cdot (-\sqrt{3}) \cdot (-1) \cdot (-\sqrt{3}) = (-3) \cdot (-1) \cdot (-\sqrt{3}) = 3 \cdot (-\sqrt{3}) = -3\sqrt{3}$$

Multiply the denominators:

$$2 \cdot 1 \cdot \sqrt{3} \cdot 2 = 4\sqrt{3}$$

Combine the numerator and denominator:

$$A = \frac{-3\sqrt{3}}{4\sqrt{3}}$$

Cancel out $\sqrt{3}$ from the numerator and denominator:

$$A = -\frac{3}{4}$$

The simplified value is:

$$-\frac{3}{4}$$