Let $x=\csc\theta$ Then the equation becomes $\sqrt{3}x^2-2(\sqrt{3}-1)x-4=0$

Solve the quadratic $\sqrt{3}x^2-2(\sqrt{3}-1)x-4=0$

Using factorization/inspection, test the roots:

• For $x=2$: $\sqrt{3}(4)-2(\sqrt{3}-1)(2)-4=4\sqrt{3}-4\sqrt{3}+4-4=0$ So, $x=2$ is a root.

Product of roots is $\frac{-4}{\sqrt{3}}$ Hence the other root is $\frac{-4/\sqrt{3}}{2}= -\frac{2}{\sqrt{3}}$

Therefore, $\csc\theta=2 \quad \text{or} \quad \csc\theta=-\frac{2}{\sqrt{3}}$

Convert to sine

Since $\csc\theta=\dfrac{1}{\sin\theta}$,

• If $\csc\theta=2$, then $\sin\theta=\frac12$

• If $\csc\theta=-\frac{2}{\sqrt{3}}$, then $\sin\theta=-\frac{\sqrt{3}}{2}$

Find all solutions in $\theta\in\left[-\frac{7\pi}{6},\frac{4\pi}{3}\right]$

Case 1: $\sin\theta=\dfrac12$

General solutions: $\theta=\frac{\pi}{6}+2k\pi \quad \text{or} \quad \theta=\frac{5\pi}{6}+2k\pi$

Now check values in the interval $\left[-\frac{7\pi}{6},\frac{4\pi}{3}\right]$:

• From $\theta=\frac{\pi}{6}+2k\pi$: $\theta=\frac{\pi}{6}$ is in the interval. $\frac{\pi}{6}-2\pi=-\frac{11\pi}{6}$ is not in the interval.

• From $\theta=\frac{5\pi}{6}+2k\pi$: $\theta=\frac{5\pi}{6}$ is in the interval. $\frac{5\pi}{6}-2\pi=-\frac{7\pi}{6}$ is also in the interval.

So this case gives 3 solutions: $\theta=-\frac{7\pi}{6},\ \frac{\pi}{6},\ \frac{5\pi}{6}$

Case 2: $\sin\theta=-\dfrac{\sqrt{3}}{2}$

General solutions: $\theta=-\frac{\pi}{3}+2k\pi \quad \text{or} \quad \theta=-\frac{2\pi}{3}+2k\pi$

Check values in the interval:

• From $\theta=-\frac{\pi}{3}+2k\pi$: $-\frac{\pi}{3}$ is in the interval, $-\frac{\pi}{3}+2\pi=\frac{5\pi}{3}$ is not, $-\frac{\pi}{3}-2\pi=-\frac{7\pi}{3}$ is not.

• From $\theta=-\frac{2\pi}{3}+2k\pi$: $-\frac{2\pi}{3}$ is in the interval, $-\frac{2\pi}{3}+2\pi=\frac{4\pi}{3}$ is in the interval, $-\frac{2\pi}{3}-2\pi=-\frac{8\pi}{3}$ is not.

So this case gives 3 solutions: $\theta=-\frac{2\pi}{3},\ -\frac{\pi}{3},\ \frac{4\pi}{3}$

Total number of solutions $3+3=6$

Compare with options The correct option is: $\boxed{\text{C: }6}$

Verification with stored answer Stored correct answer is C, which matches our result.