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To factor the trigonometric expression given:

\[
\cot^2 \alpha - \sin^2 \beta
\]

we'll proceed with the following steps:

### Step 1: Express \(\cot^2 \alpha\) in terms of sine and cosine
Using the identity for \(\cot \alpha\), we know:
\[
\cot \alpha = \frac{\cos \alpha}{\sin \alpha}
\]
Thus,
\[
\cot^2 \alpha = \frac{\cos^2 \alpha}{\sin^2 \alpha}
\]

### Step 2: Rewrite the expression
Substitute \(\cot^2 \alpha\) into the expression:
\[
\frac{\cos^2 \alpha}{\sin^2 \alpha} - \sin^2 \beta
\]

### Step 3: Check for common identities or factoring
This expression does not directly simplify through basic identities. However, if the goal is to express it in another form, like a difference of squares, we can write:
\[
\frac{\cos^2 \alpha}{\sin^2 \alpha} - \sin^2 \beta = \left(\frac{\cos \alpha}{\sin \alpha} - \sin \beta\right)\left(\frac{\cos \alpha}{\sin \alpha} + \sin \beta\right)
\]

This factors the expression as a difference of squares, yielding:

\[
\left(\frac{\cos \alpha}{\sin \alpha} - \sin \beta\right)\left(\frac{\cos \alpha}{\sin \alpha} + \sin \beta\right)
\]

### Final Answer:
\[
\cot^2 \alpha - \sin^2 \beta = \left(\frac{\cos \alpha}{\sin \alpha} - \sin \beta\right)\left(\frac{\cos \alpha}{\sin \alpha} + \sin \beta\right)
\]

Would you like a detailed explanation of any specific steps or identities used?

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Here are some follow-up questions for further practice:

1. How can you simplify expressions involving \(\sec\) and \(\csc\) functions?
2. What are the identities involving \(\sin^2 x\) and \(\cos^2 x\) that can help in factoring?
3. How does the difference of squares formula apply to trigonometric expressions?
4. How would you simplify \(\tan^2 x - \cot^2 x\)?
5. What techniques can help simplify expressions like \(\sec^2 \alpha - \csc^2 \beta\)?

**Tip:** Remember that factoring trigonometric expressions often involves rewriting terms using identities to reveal common patterns like the difference of squares.

Factor the following trigonometric expression: \( \cot^2 \alpha - \sin^2 \beta \)

Learn how to factor the trigonometric expression \( \cot^2 \alpha - \sin^2 \beta \) using the difference of squares formula. This problem explores trigonometric identities and factoring techniques, suitable for students in Grades 10-12.

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