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To solve the expression \( \sec^2{x} - 1 \) using trigonometric identities, we can apply the Pythagorean identity.

### Recall the Pythagorean identity:
\[
\sec^2{x} = 1 + \tan^2{x}
\]

So, if we substitute \( \sec^2{x} \) into the expression, we get:
\[
\sec^2{x} - 1 = (1 + \tan^2{x}) - 1
\]

Simplifying this:
\[
\sec^2{x} - 1 = \tan^2{x}
\]

Thus, the expression \( \sec^2{x} - 1 \) is equivalent to \( \tan^2{x} \).

Let me know if you need further explanation or have any questions.

Here are five follow-up questions for deeper understanding:
1. How does the Pythagorean identity relate to the unit circle?
2. What are other forms of Pythagorean identities involving sine and cosine?
3. How can you express \( \tan^2{x} \) in terms of sine and cosine?
4. What is the derivative of \( \sec^2{x} \), and how can it be useful in calculus?
5. How would the expression change if we started with \( \csc^2{x} - 1 \)?

**Tip**: When solving trigonometric identities, always try to recall and apply the most fundamental identities like the Pythagorean identities for simplification!

Using trigonometric identities, which of the following expressions is equivalent to sec²x - 1?

Learn how to simplify sec²x - 1 using the Pythagorean identity. This problem involves fundamental trigonometric identities and is suitable for students in Grades 10-12. Understand how to use the identity sec²x = 1 + tan²x to find an equivalent expression.

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