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We are tasked with finding the limit of the following expression:

\[
\lim_{{x \to \frac{\pi}{4}}} \frac{1 - \sin(2x)}{\cos^2(2x)}
\]

Let's break down the problem:

### Step 1: Simplify the trigonometric functions
- First, recall that \(\sin(2x)\) and \(\cos(2x)\) are standard double-angle trigonometric functions.

We need to evaluate the limit as \(x\) approaches \(\frac{\pi}{4}\).

### Step 2: Substitute \(x = \frac{\pi}{4}\)

At \(x = \frac{\pi}{4}\):

\[
\sin(2x) = \sin\left(2 \times \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1
\]
\[
\cos(2x) = \cos\left(2 \times \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) = 0
\]

Now, substituting into the original expression:

\[
\frac{1 - \sin(2x)}{\cos^2(2x)} = \frac{1 - 1}{\cos^2(2x)} = \frac{0}{0}
\]

This yields an indeterminate form \(\frac{0}{0}\), so we apply **L'Hôpital's Rule**.

### Step 3: Apply L'Hôpital's Rule
To apply L'Hôpital's Rule, differentiate the numerator and the denominator with respect to \(x\).

- The derivative of \(1 - \sin(2x)\) is:
  \[
  \frac{d}{dx}[1 - \sin(2x)] = -2\cos(2x)
  \]
- The derivative of \(\cos^2(2x)\) is:
  \[
  \frac{d}{dx}[\cos^2(2x)] = 2\cos(2x) \cdot (-2\sin(2x)) = -4\cos(2x)\sin(2x)
  \]

### Step 4: Evaluate the limit again
Now, the limit becomes:

\[
\lim_{{x \to \frac{\pi}{4}}} \frac{-2\cos(2x)}{-4\cos(2x)\sin(2x)}
\]

We can simplify this expression by canceling out \(\cos(2x)\) (since it's non-zero near \(x = \frac{\pi}{4}\)):

\[
\lim_{{x \to \frac{\pi}{4}}} \frac{2}{4\sin(2x)} = \frac{1}{2\sin\left(\frac{\pi}{2}\right)} = \frac{1}{2 \times 1} = \frac{1}{2}
\]

Thus, the value of the limit is \(\boxed{\frac{1}{2}}\).

### Final Answer:
The correct option is **C.** \(\frac{1}{2}\).

---

Do you want more details or have any questions?

Here are 5 related questions for further practice:
1. What is L'Hôpital's Rule, and when can it be applied?
2. How do you find the limit of \(\lim_{x \to 0} \frac{\sin(x)}{x}\)?
3. What are the double-angle formulas for sine and cosine?
4. How would the answer change if the limit were approaching a different value?
5. Can you explain the importance of recognizing indeterminate forms in calculus?

**Tip:** Always check for indeterminate forms before applying L'Hôpital's Rule!

Nilai lim x→π/4 (1 - sin(2x)) / cos^2(2x) adalah ....

Learn how to calculate the limit lim x→π/4 (1 - sin(2x)) / cos^2(2x) using trigonometric identities and L'Hôpital's Rule. This problem covers key calculus concepts and is suitable for students in Grades 10-12.

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