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The question provided involves a substitution of \( x = \sin t \) into the given integral:

\[
\int_{\frac{\sqrt{2}}{2}}^{1} \frac{\sqrt{1 - x^2}}{x^2} \, dx
\]

### Step 1: Substitution \( x = \sin t \)
Given \( x = \sin t \), we know the following relationships:
- \( dx = \cos t \, dt \)
- \( \sqrt{1 - x^2} = \cos t \) (from the Pythagorean identity \( 1 - \sin^2 t = \cos^2 t \))

### Step 2: Changing the limits of integration
When \( x = \frac{\sqrt{2}}{2} \), we need to find the corresponding \( t \):
\[
\sin t = \frac{\sqrt{2}}{2} \implies t = \frac{\pi}{4}
\]

When \( x = 1 \), we have:
\[
\sin t = 1 \implies t = \frac{\pi}{2}
\]

### Step 3: Rewriting the integral
Substitute these values into the integral:

\[
\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos t}{\sin^2 t} \cdot \cos t \, dt
\]

This simplifies to:

\[
\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos^2 t}{\sin^2 t} \, dt = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 t \, dt
\]

### Conclusion:
The integral becomes:

\[
\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 t \, dt
\]

Thus, the correct answer is:

\[
\boxed{\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 t \, dt}
\]

This corresponds to the last option in the given choices.

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Would you like further clarification on any part of the solution? Here are five related questions for deeper understanding:

1. How is the Pythagorean identity used in trigonometric substitutions?
2. What happens to the limits of integration during substitution?
3. How would the integral look if a different substitution (like \( x = \cos t \)) were made?
4. How does the cotangent function arise from the substitution process?
5. Can this integral be solved directly without substitution?

**Tip:** When dealing with trigonometric substitutions, always ensure to adjust the limits of integration according to the new variable.

После подстановки x = sin t интеграл \( \int_{\frac{\sqrt{2}}{2}}^{1} \frac{\sqrt{1 - x^2}}{x^2} \, dx \) примет вид:

Explore how the integral \( \int_{\frac{\sqrt{2}}{2}}^{1} \frac{\sqrt{1 - x^2}}{x^2} \, dx \) transforms after using the substitution \( x = \sin t \). Learn about trigonometric identities, substitution techniques, and adjusting limits in calculus.

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