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To simplify the expression 

\[
\frac{\sec^2 x - 1}{\sec x + 1},
\]

we can proceed by using trigonometric identities. Recall that:

\[
\sec^2 x - 1 = \tan^2 x.
\]

Thus, the expression becomes:

\[
\frac{\tan^2 x}{\sec x + 1}.
\]

Now, we can express \(\sec x\) in terms of \(\cos x\) and \(\tan x\) in terms of \(\sin x\) and \(\cos x\) if needed for further simplification. However, this form \(\frac{\tan^2 x}{\sec x + 1}\) is already simplified.

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**Relative Questions:**

1. How can we further simplify \(\frac{\tan^2 x}{\sec x + 1}\) using trigonometric substitution?
2. What are the key trigonometric identities used in such simplifications?
3. How does knowing the Pythagorean identities help in simplifying similar expressions?
4. Can we rewrite \(\sec x + 1\) in terms of other trigonometric functions?
5. How does simplifying trigonometric expressions assist in solving calculus problems?

**Tip:** Always check if a trigonometric expression can be simplified using basic identities before expanding or factoring further.

Simplify the expression (sec^2 x - 1) / (sec x + 1).

This guide walks through simplifying the trigonometric expression (sec^2 x - 1) / (sec x + 1) using trigonometric identities. Learn how to transform secant terms and apply the Pythagorean identity to achieve a simplified form.

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