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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.

Simplify the following trigonometric expression: (cos θ / (csc θ - 1)) * ((csc θ + 1) / (csc θ + 1))

Learn how to simplify the trigonometric expression (cos θ / (csc θ - 1)) * ((csc θ + 1) / (csc θ + 1)) by applying trigonometric identities and simplification techniques. This example covers reciprocal identities and step-by-step trigonometric simplifications. Suitable for students in Grades 10-12.

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