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 $\frac{\pi}{2}$.