Complete the table with exact trigonometric function values for 300 degrees.

1. Identify the angle and its quadrant: The angle is $300^{\circ}$. This angle is in the fourth quadrant ($270^{\circ} < 300^{\circ} < 360^{\circ}$).
2. Find the reference angle: The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. For $300^{\circ}$, the reference angle is $360^{\circ} - 300^{\circ} = 60^{\circ}$.
3. Determine the signs of trigonometric functions in the fourth quadrant: In the fourth quadrant, cosine and secant are positive, while sine, cosecant, tangent, and cotangent are negative.
4. Calculate $\cos(300^{\circ})$: The cosine of the reference angle $60^{\circ}$ is $\cos(60^{\circ}) = \frac{1}{2}$. Since $300^{\circ}$ is in the fourth quadrant, cosine is positive. Therefore, $\cos(300^{\circ}) = \frac{1}{2}$.
5. Calculate $\cot(300^{\circ})$: The cotangent of the reference angle $60^{\circ}$ is $\cot(60^{\circ}) = \frac{1}{\tan(60^{\circ})} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. Since $300^{\circ}$ is in the fourth quadrant, cotangent is negative. Therefore, $\cot(300^{\circ}) = -\frac{\sqrt{3}}{3}$.
6. Verify with given values: The table provides $\sin(300^{\circ}) = -\frac{\sqrt{3}}{2}$, $\tan(300^{\circ}) = -\sqrt{3}$, and $\sec(300^{\circ}) = 2$, and $\csc(300^{\circ}) = -\frac{2\sqrt{3}}{3}$.
- $\sin(300^{\circ})$ is negative in the 4th quadrant, and $\sin(60^{\circ}) = \frac{\sqrt{3}}{2}$, so $\sin(300^{\circ}) = -\frac{\sqrt{3}}{2}$. This matches.
- $\cos(300^{\circ})$ is positive in the 4th quadrant, and $\cos(60^{\circ}) = \frac{1}{2}$, so $\cos(300^{\circ}) = \frac{1}{2}$.
- $\tan(300^{\circ}) = \frac{\sin(300^{\circ})}{\cos(300^{\circ})} = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}$. This matches.
- $\cot(300^{\circ}) = \frac{1}{\tan(300^{\circ})} = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3}$.
- $\sec(300^{\circ}) = \frac{1}{\cos(300^{\circ})} = \frac{1}{1/2} = 2$. This matches.
- $\csc(300^{\circ}) = \frac{1}{\sin(300^{\circ})} = \frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$. This matches.