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!