Let's break down and simplify the given trigonometric expression step-by-step.

The expression to simplify is:

$\frac{\cos \theta}{\csc \theta - 1} \times \frac{\csc \theta + 1}{\csc \theta + 1}$

Step 1: Simplify the product

Since $\frac{\csc \theta + 1}{\csc \theta + 1} = 1$, this part simplifies to 1, provided $\csc \theta + 1 \neq 0$. Therefore, we can ignore this term. The simplified expression is:

$\frac{\cos \theta}{\csc \theta - 1}$

Step 2: Use the identity for $\csc \theta$

Recall that $\csc \theta = \frac{1}{\sin \theta}$. So, the expression becomes:

$\frac{\cos \theta}{\frac{1}{\sin \theta} - 1}$

Step 3: Simplify the denominator

The denominator $\frac{1}{\sin \theta} - 1$ can be rewritten with a common denominator:

$\frac{1 - \sin \theta}{\sin \theta}$

Step 4: Combine the terms

Now, the expression becomes:

$\frac{\cos \theta}{\frac{1 - \sin \theta}{\sin \theta}} = \cos \theta \times \frac{\sin \theta}{1 - \sin \theta}$

Step 5: Simplify the final expression

Now multiply $\cos \theta$ by $\frac{\sin \theta}{1 - \sin \theta}$:

$\frac{\cos \theta \sin \theta}{1 - \sin \theta}$

Thus, the simplified form of the given trigonometric expression is:

$\frac{\cos \theta \sin \theta}{1 - \sin \theta}$

Final Answer:

$\boxed{\frac{\cos \theta \sin \theta}{1 - \sin \theta}}$

Would you like a more detailed explanation on any of the steps? Let me know if you have any questions.

Here are 5 related questions to explore further:

1. What is the general identity for $\csc \theta$ and how is it used in simplifications?
2. How can other trigonometric identities, like the Pythagorean identity, simplify similar expressions?
3. What happens if $\csc \theta + 1 = 0$? What values of $\theta$ would cause this situation?
4. How can you factor trigonometric expressions involving sums or differences of sines and cosines?
5. Can you use this method to simplify other expressions involving cosecant and secant functions?

Tip: Always check for trigonometric identities, such as reciprocal and Pythagorean identities, when simplifying complex expressions.