If \(\theta \in \left[-\frac{7\pi }{6},\frac{4\pi }{3}\right]\), then the number of solutions of \(\sqrt{3}{\csc }^{2}\t…

\(\sqrt{3}{\csc }^{2}\theta −2(\sqrt{3}−1)\csc \theta −4=0\)

Let \(x=\csc ⁡\theta\).

\(\sqrt{3}{x}^{2}−2(\sqrt{3}−1)x−4=0\)

\(\sqrt{3}x(x−2)+2(x−2)=0\)

\((\sqrt{3}x+2)(x−2)=0\)

So, the solutions for \(x\) are: \(x=2 \) or \(x=-\frac{2}{\sqrt{3}}\)

Substituting \(x=\csc \theta\) back:

\(\csc \theta =2⟹\sin ⁡\theta =\frac{1}{2}\)

\(\csc \theta =−\frac{2}{\sqrt{3}}⟹\sin ⁡\theta =−\frac{\sqrt{3}}{2}\)

The solutions are: \(−\frac{7\pi }{6},−\frac{2\pi }{3},−\frac{\pi }{3},\frac{\pi }{6},\frac{5\pi }{6},\frac{4\pi }{3}\).

Therefore, the number of solutions is 6.