$\frac{3{\cos }^3\theta -2\cos \theta }{\sin \theta -3{\sin }^3\theta }$ is equal to:

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The correct answer is $\cot \theta$.

To simplify the given trigonometric expression:

$\frac{3{\cos }^3\theta -2\cos \theta }{\sin \theta -3{\sin }^3\theta }$

Step 1: Factor out common terms from the numerator and denominator.
In the numerator, factor out $\cos \theta$:
$3{\cos }^3\theta -2\cos \theta =\cos \theta (3{\cos }^2\theta -2)$

In the denominator, factor out $\sin \theta$:
$\sin \theta -3{\sin }^3\theta =\sin \theta (1-3{\sin }^2\theta )$

So, the expression becomes:

$\frac{\cos \theta (3{\cos }^2\theta -2)}{\sin \theta (1-3{\sin }^2\theta )}$

Step 2: Use the trigonometric identity ${\cos }^2\theta =1-{\sin }^2\theta$ to rewrite the numerator term.
Substitute ${\cos }^2\theta =1-{\sin }^2\theta$ into $3{\cos }^2\theta -2$:

$3(1-{\sin }^2\theta )-2=3-3{\sin }^2\theta -2=1-3{\sin }^2\theta$

Step 3: Simplify the expression.
Substitute this back into the numerator:

$\frac{\cos \theta (1-3{\sin }^2\theta )}{\sin \theta (1-3{\sin }^2\theta )}$

Cancel out the common term $(1-3{\sin }^2\theta )$ from both the numerator and denominator:

$\frac{\cos \theta }{\sin \theta }=\cot \theta$

Thus, the expression simplifies to $\cot \theta$.