Given $\theta = \frac{4\pi}{3}$, we need to find $\cos^2 \theta$.

First, determine $\cos \theta$. Since $\theta = \frac{4\pi}{3}$ is in the third quadrant, where cosine is negative, and $\cos \frac{\pi}{3} = \frac{1}{2}$:

$\cos \frac{4\pi}{3} = -\cos \frac{\pi}{3} = -\frac{1}{2}$

Next, we square this value to find $\cos^2 \theta$:

$\cos^2 \theta = \left( -\frac{1}{2} \right)^2 = \frac{1}{4}$

Thus, the exact value of $\cos^2 \theta$ is:

$\cos^2 \theta = \frac{1}{4}$

Would you like more details or have any questions? Here are some additional questions you might be interested in:

1. How do you find $\sin^2 \theta$ given $\theta = \frac{4\pi}{3}$?
2. What is the relationship between $\cos \theta$ and $\sin \theta$ for a given angle $\theta$?
3. How can the Pythagorean identity help in solving trigonometric problems?
4. What is the significance of trigonometric functions in different quadrants?
5. How can you use trigonometric identities to simplify expressions?

Tip: Squaring a trigonometric function, such as $\cos^2 \theta$, always results in a positive value since squaring any real number results in a non-negative value.