1. **State the problem:** Find the domain of the function $$g(x) = p 1 - \sin(x) \tan\left(\frac{x}{2} + \frac{1}{x}\right)$$.\n\n2. **Rewrite the problem:** We want all values of $x$ for which the expression inside the function is defined. This means we need to find where both $$\sin(x)$$ and $$\tan\left(\frac{x}{2} + \frac{1}{x}\right)$$ are defined, and where any denominators are non-zero.\n\n3. **Analyze components:** \n- $$\sin(x)$$ is defined for all real $x$.\n- For $$\tan\left(\frac{x}{2} + \frac{1}{x}\right)$$, tangent is undefined where its argument equals $$\frac{\pi}{2} + k\pi$$, where $k$ is any integer (because cosine is zero there).\n\n4. **Check the inner argument of tangent:** $$t = \frac{x}{2} + \frac{1}{x}$$ must not be equal to $$\frac{\pi}{2} + k\pi$$. Also, $x$ cannot be zero because $$\frac{1}{x}$$ is undefined at zero.\n\n5. **Domain restrictions:**\n- $$x \neq 0$$ (division by zero in $$\frac{1}{x}$$)\n- $$\frac{x}{2} + \frac{1}{x} \neq \frac{\pi}{2} + k\pi$$ for all integers $k$.\n\n6. **Summary:** The domain of $g(x)$ is all real numbers except:\n$$\boxed{x \neq 0 \quad \text{and} \quad \frac{x}{2} + \frac{1}{x} \neq \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}}$$\n\nThis domain excludes zero and all $x$ values that satisfy the equation $$\frac{x}{2} + \frac{1}{x} = \frac{\pi}{2} + k\pi$$.