Let's solve the expression provided in the image step by step.

The expression given is:

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

Step 1: Simplify the numerator

The numerator is $3 \cos^3 \theta - 2 \cos \theta$.

We can factor out $\cos \theta$ from the numerator:

$3 \cos^3 \theta - 2 \cos \theta = \cos \theta (3 \cos^2 \theta - 2)$

Step 2: Simplify the denominator

The denominator is $\sin \theta - 3 \sin^3 \theta$.

Similarly, we can factor out $\sin \theta$ from the denominator:

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

Step 3: Rewrite the expression

Now, the expression can be rewritten as:

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

Step 4: Analyze trigonometric identities

Notice that:

$\sin^2 \theta + \cos^2 \theta = 1 \quad \text{(Pythagorean identity)}$

So, $1 - 3 \sin^2 \theta = 3 \cos^2 \theta - 2$.

This means that:

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

Step 5: Simplify the expression

Since the numerator and denominator now have the same factor $3 \cos^2 \theta - 2$, and assuming this factor is non-zero, they cancel out:

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

Final Answer:

$\boxed{\cot \theta}$

This is the value of the given expression.

Would you like more details on any step, or do you have further questions?

Related Questions:

1. What is the significance of trigonometric identities in simplifying expressions?
2. How does the Pythagorean identity help in solving trigonometric equations?
3. Can you explain how to derive the cotangent identity from basic trigonometric functions?
4. What are other ways to simplify trigonometric expressions?
5. How can you use factorization in solving trigonometric equations?

Tip:

Always look for opportunities to factor expressions and apply identities in trigonometry. This often simplifies complex problems into manageable steps.