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.

Would you like more details or have any questions?

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.