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We are tasked with finding the following limit:

\[
\lim_{{x \to 0}} \left( 1 - \tan^2(x) \right)^{\frac{1}{\sin^2(2x)}}
\]

### Step 1: Analyze the behavior of \(\tan^2(x)\) as \(x \to 0\)

We know that \(\tan(x) \approx x\) when \(x\) is small, so:

\[
\tan^2(x) \approx x^2 \quad \text{as} \quad x \to 0
\]

Thus, \(1 - \tan^2(x)\) behaves as:

\[
1 - \tan^2(x) \approx 1 - x^2 \quad \text{as} \quad x \to 0
\]

### Step 2: Analyze the behavior of \(\sin^2(2x)\) as \(x \to 0\)

Similarly, for small \(x\), \(\sin(2x) \approx 2x\), so:

\[
\sin^2(2x) \approx (2x)^2 = 4x^2 \quad \text{as} \quad x \to 0
\]

### Step 3: Transform the expression

Now we have the limit:

\[
\lim_{{x \to 0}} \left( 1 - x^2 \right)^{\frac{1}{4x^2}}
\]

### Step 4: Apply logarithms for simplification

To simplify the expression further, let's take the natural logarithm of the limit expression. Define:

\[
L = \lim_{{x \to 0}} \left( 1 - x^2 \right)^{\frac{1}{4x^2}}
\]

Taking the natural logarithm of both sides:

\[
\ln(L) = \lim_{{x \to 0}} \frac{1}{4x^2} \ln(1 - x^2)
\]

For small \(x\), we can use the approximation \(\ln(1 - x^2) \approx -x^2\), so:

\[
\ln(L) = \lim_{{x \to 0}} \frac{1}{4x^2} (-x^2) = \lim_{{x \to 0}} \frac{-x^2}{4x^2} = -\frac{1}{4}
\]

### Step 5: Exponentiate the result

Now, exponentiating both sides, we get:

\[
L = e^{-\frac{1}{4}}
\]

Thus, the value of the original limit is:

\[
\boxed{e^{-\frac{1}{4}}}
\]

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do approximations for small \(x\) like \(\tan(x) \approx x\) help in solving limits?
2. Can you derive the approximation \(\ln(1 - x^2) \approx -x^2\) using a Taylor series expansion?
3. What is the significance of logarithmic transformations in limit problems?
4. How would the result change if the exponent in the original problem was \(\frac{1}{\sin(2x)}\) instead of \(\frac{1}{\sin^2(2x)}\)?
5. What would the result be if we consider a different trigonometric function, like \(\cos^2(x)\), in place of \(\tan^2(x)\)?

**Tip:** Logarithms are a powerful tool for simplifying limits that involve exponential forms.

Learn how to solve the limit problem lim (x→0) (1 - tan^2(x))^(1/sin^2(2x)). This problem explores key concepts such as trigonometric approximations and logarithmic transformations. Suitable for students in undergraduate calculus.

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