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To solve the integral 

\[
\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx,
\]

we can use a trigonometric identity to simplify \(\cos^2 x\). 

### Step 1: Use the Power-Reduction Formula
The identity for \(\cos^2 x\) is:
\[
\cos^2 x = \frac{1 + \cos(2x)}{2}.
\]

### Step 2: Substitute the Identity into the Integral
This transforms the integral into:
\[
\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \int_{0}^{\frac{\pi}{2}} \frac{1 + \cos(2x)}{2} \, dx.
\]

We can factor out the \(\frac{1}{2}\):
\[
= \frac{1}{2} \int_{0}^{\frac{\pi}{2}} (1 + \cos(2x)) \, dx.
\]

### Step 3: Split the Integral
Now, split the integral:
\[
= \frac{1}{2} \left( \int_{0}^{\frac{\pi}{2}} 1 \, dx + \int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx \right).
\]

### Step 4: Evaluate Each Integral Separately
1. **First Integral**:
   \[
   \int_{0}^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2}.
   \]

2. **Second Integral**:
   For \(\int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx\), use substitution. Let \(u = 2x\), so \(du = 2 \, dx\) or \(dx = \frac{du}{2}\).

   Changing the limits:
   - When \(x = 0\), \(u = 0\).
   - When \(x = \frac{\pi}{2}\), \(u = \pi\).

   The integral becomes:
   \[
   \int_{0}^{\frac{\pi}{2}} \cos(2x) \, dx = \int_{0}^{\pi} \cos(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{\pi} \cos(u) \, du.
   \]

   Now, integrate \(\cos(u)\):
   \[
   \frac{1}{2} \int_{0}^{\pi} \cos(u) \, du = \frac{1}{2} \left[ \sin(u) \right]_{0}^{\pi} = \frac{1}{2} (0 - 0) = 0.
   \]

So, the second integral is zero.

### Step 5: Combine Results
The integral now simplifies to:
\[
\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}.
\]

### Final Answer
\[
\int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx = \frac{\pi}{4}.
\]

Would you like more details on any specific part of the solution?

### Related Questions
1. How do we derive the power-reduction formula for trigonometric functions?
2. Can we solve similar integrals for \(\sin^2 x\) over the same interval?
3. What is the significance of using substitution in integrals with trigonometric functions?
4. How would the result change if the limits of integration were different?
5. What other techniques can we use to solve integrals involving \(\cos^2 x\)?

### Tip
For integrals involving squares of trigonometric functions, the power-reduction identities are particularly useful to simplify the expression before integrating.

Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx \)

Learn how to solve the definite integral \( \int_{0}^{\frac{\pi}{2}} \cos^2 x \, dx \) using trigonometric identities and integration techniques. This solution covers the power-reduction formula and detailed steps for simplifying and evaluating the integral. Suitable for advanced high school students in Grades 11-12.

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