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To simplify the given trigonometric expression:

\[
\text{tan}^2 x - \text{sec}^2 x
\]

We can use the Pythagorean identity:

\[
\text{sec}^2 x = 1 + \text{tan}^2 x
\]

Substitute this identity into the original expression:

\[
\text{tan}^2 x - \text{sec}^2 x = \text{tan}^2 x - (1 + \text{tan}^2 x)
\]

Simplifying further:

\[
\text{tan}^2 x - 1 - \text{tan}^2 x = -1
\]

Thus, the simplified expression is:

\[
\boxed{-1}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What are the other Pythagorean trigonometric identities?
2. How do you derive the identity \(\sec^2 x = 1 + \tan^2 x\)?
3. Can this expression have a real value if \(x\) is undefined for secant?
4. How does the tangent and secant function relate geometrically on the unit circle?
5. What is the domain of \(\sec(x)\) and \(\tan(x)\)?

**Tip:** When simplifying trigonometric expressions, always look for identities like Pythagorean or angle sum formulas to reduce complexity.

Simplify the trigonometric expression: tan^2(x) - sec^2(x)

Simplify the expression tan^2(x) - sec^2(x) using Pythagorean trigonometric identities. This problem demonstrates the use of key formulas such as sec^2(x) = 1 + tan^2(x), making it ideal for students in Grades 10-12.

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