We are tasked with simplifying the expression:

$(\csc(\theta) - 1)(\csc(\theta) + 1)$

This is a product of two binomials, and we can simplify it using the difference of squares formula. The difference of squares states that:

$(a - b)(a + b) = a^2 - b^2$

Here, $a = \csc(\theta)$ and $b = 1$, so applying the formula gives:

$(\csc(\theta) - 1)(\csc(\theta) + 1) = \csc^2(\theta) - 1$

Now, recall the Pythagorean identity for trigonometric functions:

$\csc^2(\theta) = 1 + \cot^2(\theta)$

Therefore, we can replace $\csc^2(\theta)$ with $1 + \cot^2(\theta)$ in the expression:

$\csc^2(\theta) - 1 = (1 + \cot^2(\theta)) - 1$

Simplifying the right-hand side:

$\cot^2(\theta)$

Thus, the simplified form of the expression is:

$\cot^2(\theta)$

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Would you like more details on any step, or have other questions?

Here are some related questions:

1. What is the Pythagorean identity for $\csc(\theta)$ and $\cot(\theta)$?
2. How do you derive the identity $\csc^2(\theta) = 1 + \cot^2(\theta)$?
3. Can you simplify expressions involving other trigonometric identities, like $\sec$ and $\tan$?
4. What are some common simplification strategies for trigonometric expressions?
5. How can you apply trigonometric identities to solve problems in calculus?

Tip: When simplifying trigonometric expressions, always check for opportunities to apply Pythagorean identities to reduce terms to more manageable forms!