Let's simplify the given expression:

\[
\frac{\csc^2(-\pi/4)}{-8\sin(-\pi/4)}
\]

### Step-by-Step Solution:

1. **Evaluate \(\csc^2(-\pi/4)\):**
   - Recall that \(\csc(x) = \frac{1}{\sin(x)}\). 
   - \(\sin(-\pi/4) = -\sin(\pi/4) = -\frac{\sqrt{2}}{2}\).
   - Therefore, \(\csc(-\pi/4) = \frac{1}{\sin(-\pi/4)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}\).
   - Simplify \(-\frac{2}{\sqrt{2}}\) to \(-\sqrt{2}\).
   - So, \(\csc^2(-\pi/4) = (-\sqrt{2})^2 = 2\).

2. **Evaluate \(\sin(-\pi/4)\):**
   - As noted, \(\sin(-\pi/4) = -\frac{\sqrt{2}}{2}\).

3. **Simplify the denominator \(-8\sin(-\pi/4)\):**
   \[
   -8\sin(-\pi/4) = -8 \cdot \left(-\frac{\sqrt{2}}{2}\right) = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2}
   \]

4. **Put it all together:**
   \[
   \frac{\csc^2(-\pi/4)}{-8\sin(-\pi/4)} = \frac{2}{4\sqrt{2}}
   \]

5. **Simplify the fraction:**
   \[
   \frac{2}{4\sqrt{2}} = \frac{1}{2\sqrt{2}}
   \]

6. **Rationalize the denominator:**
   \[
   \frac{1}{2\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{4}
   \]

### Final Answer:
\[
\boxed{\frac{\sqrt{2}}{4}}
\]

Would you like any further details or have questions on any steps?

### Related Questions:
1. How would you simplify \(\csc^2(\pi/4)\) without using negative angles?
2. Can you explain why \(\sin(-x) = -\sin(x)\)?
3. How do you rationalize a denominator, and why is it necessary?
4. What is the general process for simplifying trigonometric expressions?
5. How do the even-odd identities affect trigonometric functions?

### Tip:
When working with trigonometric functions of negative angles, remember that \(\sin(-x) = -\sin(x)\) and \(\csc(-x) = -\csc(x)\) because sine is an odd function.

Explore the step-by-step solution to simplify the trigonometric expression (csc^2(-pi/4)) / (-8sin(-pi/4)) using key trigonometric identities and concepts like csc(x) = 1/sin(x) and the odd function property of sine. Suitable for high school students in Grades 10-12.

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