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The expression provided appears to involve trigonometric identities, particularly with cotangent (\(\cot\)) and cosecant (\(\csc\)) functions.

Let's break it down step by step:

### Expression Breakdown:
1. The first part of the expression is:
   \[
   (\cot A + \csc A) - (\csc A + \cot A)(\csc A - \cot A)/(1 - \csc A + \cot A)
   \]
   - This involves a combination of cotangent and cosecant of angle \(A\).
   - There is a subtraction of two terms, and one term has a difference of squares type structure \((\csc A + \cot A)(\csc A - \cot A)\), which simplifies using the difference of squares identity.

2. **Difference of Squares**:
   \[
   (\csc A + \cot A)(\csc A - \cot A) = \csc^2 A - \cot^2 A
   \]
   So the expression now becomes:
   \[
   (\cot A + \csc A) - (\csc^2 A - \cot^2 A)/(1 - \csc A + \cot A)
   \]

3. **Next Steps**:
   The next step appears to simplify the terms inside the expression. The focus should be on simplifying the fraction.

### Further Simplification:
The expression is simplified as follows:
\[
= (\cot A + \csc A)(1 - \csc A + \cot A)/(1 - \csc A + \cot A)
\]
In this step, the common denominator \((1 - \csc A + \cot A)\) cancels out, leaving:
\[
= \cot A + \csc A
\]

### Conclusion:
The final simplified expression is:
\[
\cot A + \csc A
\]
This shows that after simplifying the entire expression, we end up back with the original \(\cot A + \csc A\).

### Summary:
- The provided expression is simplified using the difference of squares identity.
- The terms in the numerator and denominator cancel out, leaving the original \(\cot A + \csc A\).

Would you like further details or clarification on any part of this explanation?

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Here are 5 related questions that might interest you:
1. How do the identities for cotangent and cosecant functions help in simplifying expressions?
2. What is the difference of squares identity and how is it used in trigonometry?
3. How can we derive the Pythagorean identities from the trigonometric functions used here?
4. What other trigonometric identities simplify similar expressions involving cotangent and cosecant?
5. How do the cancellation of terms in complex trigonometric fractions work?

**Tip**: Always look for factoring opportunities when simplifying trigonometric expressions; this often reveals cancellations or simplifications that are not immediately obvious.

Simplify the expression involving cotangent and cosecant: (cot A + csc A) - (csc A + cot A)(csc A - cot A)/(1 - csc A + cot A).

Learn how to simplify a trigonometric expression involving cotangent and cosecant using the difference of squares identity. This problem includes the expression (cot A + csc A) - (csc A + cot A)(csc A - cot A)/(1 - csc A + cot A) and its step-by-step solution.

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