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The problem asks to evaluate the trigonometric limit:

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

### Solution:
To solve this, we will apply a change of variable and use limit properties.

#### Step 1: Change of Variable
Let us define a new variable \( u \) such that:
\[
u = x - \frac{\pi}{2}
\]
Thus, when \( x \to \frac{\pi}{2} \), it follows that \( u \to 0 \).

Now, replace \( x \) in terms of \( u \):
\[
x = u + \frac{\pi}{2}
\]
Thus, the limit becomes:
\[
\lim_{{u \to 0}} \frac{1 - \sin\left(u + \frac{\pi}{2}\right)}{u}
\]

#### Step 2: Simplify the Expression
Using the trigonometric identity \( \sin\left(\frac{\pi}{2} + u\right) = \cos(u) \), the expression becomes:
\[
\lim_{{u \to 0}} \frac{1 - \cos(u)}{u}
\]

#### Step 3: Apply Known Limit
We know the standard limit:
\[
\lim_{{u \to 0}} \frac{1 - \cos(u)}{u} = 0
\]

Thus, the result of the limit is:
\[
\boxed{0}
\]

### Final Answer:
The value of the limit is 0.

---

Would you like further details or have any questions? Here are some related questions to expand on this topic:

1. What other trigonometric limits are commonly used in calculus?
2. How can L'Hôpital's Rule be applied to solve limits involving trigonometric functions?
3. What are the key trigonometric identities that simplify limit problems?
4. How can Taylor series expansions help in evaluating trigonometric limits?
5. What is the significance of the small-angle approximation in trigonometric limits?

**Tip:** When facing trigonometric limits, always consider applying trigonometric identities or small-angle approximations to simplify the expression.

Evaluar, haciendo cambio de variable, el límite trigonométrico que se presenta a continuación: 

lim_{x → π/2} (1 - sin(x)) / (x - π/2)

Learn how to evaluate the trigonometric limit lim_{x → π/2} (1 - sin(x)) / (x - π/2) by applying a change of variable and using limit properties. This detailed solution covers key concepts in calculus, including trigonometric identities and limit theorems.

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