If $\csc \theta +\cot \theta =\sqrt{3}$, find the value of $\csc \theta$.

We know the standard trigonometric identity:
$${\csc }^2\theta -{\cot }^2\theta =1$$

$$(\csc \theta -\cot \theta )(\csc \theta +\cot \theta )=1$$

Given that $\csc \theta +\cot \theta =\sqrt{3}$, we can substitute it into the identity equation:

$$\csc \theta -\cot \theta =\frac{1}{\csc \theta +\cot \theta }=\frac{1}{\sqrt{3}}$$

Now add the two systems of equations together:

$$(\csc \theta +\cot \theta )+(\csc \theta -\cot \theta )=\sqrt{3}+\frac{1}{\sqrt{3}}$$

$$2\csc \theta =\frac{3+1}{\sqrt{3}}=\frac{4}{\sqrt{3}}$$

$$\csc \theta =\frac{2}{\sqrt{3}}$$