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Let's solve the given limit problem.

The function is defined as:

\[
f(x) = \frac{\sin x - 1}{\cos^2 x - 1}
\]

We are tasked with evaluating:

\[
\lim_{x \to \frac{\pi}{2}} f(x)
\]

First, simplify the expression for \( f(x) \).

### Step 1: Simplify the denominator

Notice that the denominator \( \cos^2 x - 1 \) can be written using the identity \( \cos^2 x - 1 = -\sin^2 x \). So, we rewrite \( f(x) \) as:

\[
f(x) = \frac{\sin x - 1}{-\sin^2 x}
\]

### Step 2: Apply the limit

We now want to find the limit as \( x \to \frac{\pi}{2} \):

\[
\lim_{x \to \frac{\pi}{2}} \frac{\sin x - 1}{-\sin^2 x}
\]

At \( x = \frac{\pi}{2} \), we have \( \sin\left(\frac{\pi}{2}\right) = 1 \). Substituting into the expression:

\[
f\left(\frac{\pi}{2}\right) = \frac{1 - 1}{-1^2} = \frac{0}{-1} = 0
\]

So the limit of \( f(x) \) as \( x \to \frac{\pi}{2} \) is \( 0 \).

### Step 3: Compare with the choices

- **Option A:** \(\lim_{x \to \frac{\pi}{2}} \frac{-1}{1 + \sin x}\) — This does not yield \( 0 \).
- **Option B:** \(\lim_{x \to \frac{\pi}{2}} \frac{\sin x - 1}{1 + \sin^2 x}\) — This also does not simplify to \( 0 \).
- **Option C:** \(\lim_{x \to \frac{\pi}{2}} \sec x\) — This does not simplify to \( 0 \).
- **Option D:** \(\lim_{x \to \frac{\pi}{2}} (\tan x - \sec x)\) — Let’s check this one.

At \( x = \frac{\pi}{2} \), we have \( \tan\left(\frac{\pi}{2}\right) \to \infty \) and \( \sec\left(\frac{\pi}{2}\right) \to \infty \), but their difference approaches \( 0 \).

Thus, the correct option is **D**.

Would you like more details on this solution or help with another question?

Here are five related questions:
1. How can trigonometric identities simplify limit expressions?
2. What is L'Hôpital's Rule, and when can it be applied?
3. How do you handle indeterminate forms like \( \frac{0}{0} \) in limits?
4. What are common limits involving \( \sin(x) \) and \( \cos(x) \)?
5. How do limits behave when approaching \( \frac{\pi}{2} \) for functions like \( \tan(x) \)?

**Tip:** Always check for trigonometric identities that simplify the expressions before calculating limits.

If f(x) = (sin(x) - 1) / (cos^2(x) - 1), then lim x→π/2 f(x) is equivalent to which of the following?

Solve the limit problem lim x→π/2 for the function f(x) = (sin(x) - 1) / (cos^2(x) - 1) by applying trigonometric identities and simplifications. This example explores key limit concepts and trigonometric function behaviors.

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