1. Write the determinant

We need

$$\left| \begin{matrix} 0 & \cos x & -\sin x\\ \sin x & 0 & \cos x\\ \cos x & \sin x & 0 \end{matrix} \right|=0.$$

Let $s=\sin x,\qquad c=\cos x.$ Then the determinant becomes

$$\left| \begin{matrix} 0 & c & -s\\ s & 0 & c\\ c & s & 0 \end{matrix} \right|.$$

2. Evaluate the determinant

Expanding along the first row,

$$D=0\cdot\left|\begin{matrix}0&c\\ s&0\end{matrix}\right|-c\cdot\left|\begin{matrix}s&c\\ c&0\end{matrix}\right|+(-s)\cdot\left|\begin{matrix}s&0\\ c&s\end{matrix}\right|.$$

Now,

$$\left|\begin{matrix}s&c\\ c&0\end{matrix}\right|=s\cdot 0-c\cdot c=-c^2,$$

so the second term is

$$-c(-c^2)=c^3.$$

Also,

$$\left|\begin{matrix}s&0\\ c&s\end{matrix}\right|=s\cdot s-0\cdot c=s^2,$$

so the third term is

$$(-s)(s^2)=-s^3.$$

Hence

$$D=c^3-s^3.$$

Condition $D=0$ gives

$$c^3-s^3=0 \implies c^3=s^3 \implies c=s.$$

3. Solve $\sin x=\cos x$ on $[0,2\pi]$

$$\tan x=1 \implies x=\frac{\pi}{4}+n\pi.$$

In $[0,2\pi]$, the solutions are

$$x=\frac{\pi}{4},\ \frac{5\pi}{4}.$$

So

$$S=\left\{\frac{\pi}{4},\frac{5\pi}{4}\right\}.$$

4. Compute the required sum

We need

$$\sum_{x\in S}\tan\left(\frac{\pi}{3}+x\right) =\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right)+\tan\left(\frac{\pi}{3}+\frac{5\pi}{4}\right).$$

Since tangent has period $\pi$,

$$\tan\left(\frac{\pi}{3}+\frac{5\pi}{4}\right) =\tan\left(\frac{\pi}{3}+\frac{\pi}{4}+\pi\right) =\tan\left(\frac{7\pi}{12}\right).$$

Thus both terms are equal, so

$$\text{sum}=2\tan\left(\frac{7\pi}{12}\right).$$

Now,

$$\tan\left(\frac{7\pi}{12}\right)=\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) =\frac{\tan\frac{\pi}{3}+\tan\frac{\pi}{4}}{1-\tan\frac{\pi}{3}\tan\frac{\pi}{4}} =\frac{\sqrt3+1}{1-\sqrt3}.$$

Rationalizing,

$$\frac{\sqrt3+1}{1-\sqrt3}\cdot\frac{1+\sqrt3}{1+\sqrt3} =\frac{(\sqrt3+1)^2}{1-3} =\frac{3+1+2\sqrt3}{-2} =-2-\sqrt3.$$

Therefore,

$$\text{sum}=2(-2-\sqrt3)=-4-2\sqrt3.$$

5. Compare with options

This matches $\boxed{-4-2\sqrt3}$ which is Option D.

6. Comparison with stored correct answer

Stored correct answer is C: $-2-\sqrt3$, but our computed sum is $-4-2\sqrt3.$ So the stored answer appears to be incorrect.