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Let's simplify the given expression step by step. The expression is:

\[
\frac{\sin(\theta - \pi) \cdot \cot \left(\theta - \frac{7\pi}{2} \right) \cdot \sec(\theta - \pi)}{\cos \left( \theta - \frac{3\pi}{2} \right) \cdot \csc\left(\theta + \frac{5\pi}{2}\right)}
\]

### Step 1: Use Trigonometric Identities for Angle Subtraction
We will simplify the trigonometric functions for each term based on standard angle subtraction formulas:

1. **\(\sin(\theta - \pi)\):**
   \[
   \sin(\theta - \pi) = -\sin(\theta)
   \]

2. **\(\cot\left(\theta - \frac{7\pi}{2}\right)\):**
   \[
   \cot\left(\theta - \frac{7\pi}{2}\right) = \cot\left(\theta + \frac{\pi}{2}\right) = -\tan(\theta)
   \]
   This is because \(\cot\left(\theta + \frac{\pi}{2}\right) = -\tan(\theta)\).

3. **\(\sec(\theta - \pi)\):**
   \[
   \sec(\theta - \pi) = -\sec(\theta)
   \]

4. **\(\cos\left(\theta - \frac{3\pi}{2}\right)\):**
   \[
   \cos\left(\theta - \frac{3\pi}{2}\right) = \sin(\theta)
   \]

5. **\(\csc\left(\theta + \frac{5\pi}{2}\right)\):**
   \[
   \csc\left(\theta + \frac{5\pi}{2}\right) = \csc\left(\theta + \frac{\pi}{2}\right) = \sec(\theta)
   \]

### Step 2: Substitution into the Original Expression
Substitute these simplified trigonometric expressions into the original expression:

\[
\frac{-\sin(\theta) \cdot (-\tan(\theta)) \cdot (-\sec(\theta))}{\sin(\theta) \cdot \sec(\theta)}
\]

### Step 3: Simplify the Expression
Now simplify the entire expression:

\[
= \frac{-\sin(\theta) \cdot \tan(\theta) \cdot \sec(\theta)}{\sin(\theta) \cdot \sec(\theta)}
\]

Since \(\sec(\theta)\) cancels out from both numerator and denominator, and \(\sin(\theta)\) also cancels out:

\[
= -\tan(\theta)
\]

### Final Answer:
The simplified form of the given expression is:

\[
-\tan(\theta)
\]

Let me know if you would like more details on any step!

### Relative Questions:
1. What are some other common angle subtraction identities for sine and cosine?
2. How do you simplify \(\sec\) and \(\csc\) functions in complex expressions?
3. What is the significance of using trigonometric identities in simplifying expressions?
4. Can you explain how the identity \(\cot(\theta + \frac{\pi}{2}) = -\tan(\theta)\) is derived?
5. How can we apply these principles in calculus, for example in derivatives involving trigonometric functions?

### Tip:
Always check for symmetry in trigonometric functions when simplifying complex expressions involving multiples of \(\pi\) or \(\frac{\pi}{2}\).

Simplify the trigonometric expression: sin(θ - π) * cot(θ - 7π/2) * sec(θ - π) / [cos(θ - 3π/2) * csc(θ + 5π/2)]

Learn how to simplify a complex trigonometric expression involving sine, cosine, cotangent, secant, and cosecant using angle subtraction identities. This problem provides a step-by-step solution, suitable for high school students in Grades 11-12.

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