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We are tasked with proving the trigonometric identity:

\[
(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1
\]

### Step 1: Recognize the Difference of Squares Formula
The given expression is in the form:

\[
(a + b)(a - b) = a^2 - b^2
\]

In this case:
- \(a = \sec \theta\)
- \(b = \tan \theta\)

Thus, applying the difference of squares formula:

\[
(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \sec^2 \theta - \tan^2 \theta
\]

### Step 2: Use Trigonometric Identity
We know from trigonometric identities that:

\[
\sec^2 \theta - \tan^2 \theta = 1
\]

This follows from the Pythagorean identity for secant and tangent:

\[
\sec^2 \theta = 1 + \tan^2 \theta
\]

Thus:

\[
\sec^2 \theta - \tan^2 \theta = (1 + \tan^2 \theta) - \tan^2 \theta = 1
\]

### Conclusion:
The original expression simplifies to 1, and the identity is established.

Would you like further details or clarification?

Here are some related questions to expand your understanding:
1. How does the difference of squares formula apply to trigonometric identities?
2. What is the Pythagorean identity involving secant and tangent?
3. How would this identity change for sine and cosine?
4. What are other common identities that involve secant and tangent?
5. Can you derive the Pythagorean identity from the unit circle?

**Tip**: Knowing the basic Pythagorean identities can simplify solving complex trigonometric expressions.

(sec θ + tan θ)(sec θ - tan θ) = 1. Multiply and write the left-side expression as the difference of two squares.

Learn how to establish the identity (sec θ + tan θ)(sec θ - tan θ) = 1 by applying the difference of squares and using the Pythagorean identity. Suitable for students in Grades 10-12.

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