The expression \(-\ln(\csch(x) + \coth(x))\) can be simplified as follows:

First, recall the definitions of \(\csch(x)\) and \(\coth(x)\):
- \(\csch(x) = \frac{1}{\sinh(x)}\)
- \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\)

Thus,
\[
\csch(x) + \coth(x) = \frac{1}{\sinh(x)} + \frac{\cosh(x)}{\sinh(x)} = \frac{1 + \cosh(x)}{\sinh(x)}
\]

Now, substitute this into the logarithmic expression:
\[
-\ln\left(\csch(x) + \coth(x)\right) = -\ln\left(\frac{1 + \cosh(x)}{\sinh(x)}\right)
\]

We can split the logarithm:
\[
-\ln\left(\frac{1 + \cosh(x)}{\sinh(x)}\right) = -\left(\ln(1 + \cosh(x)) - \ln(\sinh(x))\right)
\]

This simplifies to:
\[
-\ln(1 + \cosh(x)) + \ln(\sinh(x))
\]

Thus, the final simplified form of the expression is:
\[
\ln(\sinh(x)) - \ln(1 + \cosh(x))
\]

Let me know if you'd like more details or have any questions!

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Here are 5 questions to consider:
1. What are the definitions of hyperbolic functions like \(\sinh(x)\) and \(\cosh(x)\)?
2. How do the logarithmic properties apply to this problem?
3. Can you express \(\cosh(x)\) and \(\sinh(x)\) in terms of exponentials?
4. What would the expression look like for large values of \(x\)?
5. How can we interpret this result geometrically?

**Tip**: Remember, hyperbolic identities are similar to trigonometric identities but for hyperbolic functions!

This detailed solution explains how to simplify the expression −ln(csch(x) + coth(x)) using hyperbolic identities and logarithmic properties. Learn key mathematical concepts such as hyperbolic functions and logarithmic simplification. Suitable for advanced high school or college-level students.

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