The problem asks to evaluate the expression $\cos(\frac{4\pi}{3}) - \sin(\frac{4\pi}{3})$.

1. Problem Description

The problem asks to evaluate the expression $\cos(\frac{4\pi}{3}) - \sin(\frac{4\pi}{3})$.

2. Solution Steps

We need to find the values of $\cos(\frac{4\pi}{3})$ and $\sin(\frac{4\pi}{3})$.

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

We can write $\frac{4\pi}{3} = \pi + \frac{\pi}{3}$.

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

$\sin(\frac{4\pi}{3}) = \sin(\pi + \frac{\pi}{3}) = -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$

Now we substitute these values into the expression:

$\cos(\frac{4\pi}{3}) - \sin(\frac{4\pi}{3}) = -\frac{1}{2} - (-\frac{\sqrt{3}}{2}) = -\frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{3} - 1}{2}$

3. Final Answer

The final answer is $\frac{\sqrt{3} - 1}{2}$.