1. The problem is to find the value of $\cot(x + \frac{\pi}{3})$ or express it in terms of $x$. 2. Recall the cotangent addition formula: $$\cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}$$ where $A = x$ and $B = \frac{\pi}{3}$. 3. We know that: $$\cot \frac{\pi}{3} = \frac{1}{\tan \frac{\pi}{3}} = \frac{1}{\sqrt{3}}$$ 4. Substitute into the formula: $$\cot(x + \frac{\pi}{3}) = \frac{\cot x \cdot \frac{1}{\sqrt{3}} - 1}{\cot x + \frac{1}{\sqrt{3}}}$$ 5. To simplify, multiply numerator and denominator by $\sqrt{3}$: $$= \frac{\cot x - \sqrt{3}}{\sqrt{3} \cot x + 1}$$ 6. This expression gives $\cot(x + \frac{\pi}{3})$ in terms of $\cot x$. Final answer: $$\boxed{\cot(x + \frac{\pi}{3}) = \frac{\cot x - \sqrt{3}}{\sqrt{3} \cot x + 1}}$$