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Let's break down and establish the trigonometric identity given in the problem:

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

### Step 1: Use the Difference of Squares
The left-hand side is in the form of a product of two binomials, which is the **difference of squares** formula:
\[
(a + b)(a - b) = a^2 - b^2
\]
Here, \(a = 1\) and \(b = \csc \theta\), so applying the formula:
\[
(1 + \csc \theta)(1 - \csc \theta) = 1^2 - (\csc \theta)^2 = 1 - \csc^2 \theta
\]

### Step 2: Simplify using a Trigonometric Identity
We know from trigonometric identities that:
\[
1 - \csc^2 \theta = -\cot^2 \theta
\]

This is a standard identity derived from the Pythagorean identity for cotangent and cosecant:
\[
\csc^2 \theta - \cot^2 \theta = 1 \quad \Rightarrow \quad 1 - \csc^2 \theta = -\cot^2 \theta
\]

### Conclusion
Thus, we have established that:
\[
(1 + \csc \theta)(1 - \csc \theta) = -\cot^2 \theta
\]
which proves the identity.

Do you want more details or further explanation on this?

Here are 5 related questions to explore further:
1. What is the Pythagorean identity involving \(\csc \theta\) and \(\cot \theta\)?
2. How is the difference of squares formula applied in other trigonometric identities?
3. Can you derive \(\csc^2 \theta - \cot^2 \theta = 1\) from basic definitions?
4. What are other common trigonometric identities involving \(\csc \theta\) and \(\cot \theta\)?
5. How would this identity change for other values of \(\theta\) in different quadrants?

**Tip:** Always look for opportunities to apply algebraic identities like the difference of squares to simplify complex trigonometric expressions.

(1 + csc θ)(1 - csc θ) = -cot^2 θ. Multiply and write the left side expression as the difference of two squares.

Learn how to prove the trigonometric identity (1 + csc θ)(1 - csc θ) = -cot^2 θ using the difference of squares and Pythagorean identity. This step-by-step solution is ideal for high school students (Grades 10-12).

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