https://r2.erweima.ai/mathbot/image/1726061035445z7c8otd7.jpg

The image contains the mathematical limit expression:

\[
\lim_{x \to \frac{\pi}{2}} \frac{1 - \cos(x)}{2 \sec(x)}
\]

Let's solve this step-by-step.

### Step 1: Simplify the expression

First, recall the identity for secant:

\[
\sec(x) = \frac{1}{\cos(x)}
\]

Substituting this into the given expression:

\[
\frac{1 - \cos(x)}{2 \sec(x)} = \frac{1 - \cos(x)}{2 \cdot \frac{1}{\cos(x)}} = \frac{(1 - \cos(x)) \cdot \cos(x)}{2}
\]

So the expression becomes:

\[
\frac{\cos(x) - \cos^2(x)}{2}
\]

### Step 2: Evaluate the limit as \( x \to \frac{\pi}{2} \)

At \( x = \frac{\pi}{2} \):

\[
\cos\left(\frac{\pi}{2}\right) = 0
\]

Thus, when we substitute \( x = \frac{\pi}{2} \) into the simplified expression:

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

Therefore, the limit is:

\[
\lim_{x \to \frac{\pi}{2}} \frac{1 - \cos(x)}{2 \sec(x)} = 0
\]

### Conclusion:
The value of the limit is 0.

---

Do you want more details or have any questions? Here are some related questions:

1. What other trigonometric identities can be useful in limits involving cosine and secant?
2. How do we handle indeterminate forms in limits?
3. What is L'Hopital's Rule and when is it applied in limit problems?
4. Can you explain the difference between a limit approaching 0 versus approaching infinity?
5. How would the result change if the limit were approaching a different value, like \( \pi \) instead of \( \frac{\pi}{2} \)?

**Tip:** Always check for trigonometric identities that can simplify limit problems before using advanced techniques like L'Hopital's Rule.

What is the limit of (1 - cos(x)) / (2 * sec(x)) as x approaches π/2?

Solve the trigonometric limit of (1 - cos(x)) / (2 * sec(x)) as x approaches π/2. This solution involves key trigonometric identities and limit theorems, suitable for undergraduate calculus students.

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