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es sobre las transformadas de laplace, hay que resolverlo aplicando propiedades

To solve the Laplace transform of the given function \( f(t) = \cos^2(2t) \), we can approach the problem using standard properties of the Laplace transform.

### Step-by-Step Solution:

1. **Use a Trigonometric Identity**:
   The expression \( \cos^2(2t) \) can be simplified using the identity:
   \[
   \cos^2(x) = \frac{1 + \cos(2x)}{2}
   \]
   Substituting \( x = 2t \), we get:
   \[
   \cos^2(2t) = \frac{1 + \cos(4t)}{2}
   \]
   
   So, \( f(t) \) becomes:
   \[
   f(t) = \frac{1 + \cos(4t)}{2}
   \]

2. **Break the Function into Simpler Terms**:
   Now we have:
   \[
   f(t) = \frac{1}{2} + \frac{\cos(4t)}{2}
   \]

3. **Apply the Laplace Transform**:
   We now take the Laplace transform of each term separately:
   
   - The Laplace transform of a constant \( a \) is:
     \[
     \mathcal{L}\{a\} = \frac{a}{s}
     \]
     So, for \( \frac{1}{2} \), the Laplace transform is:
     \[
     \mathcal{L}\left\{\frac{1}{2}\right\} = \frac{1}{2s}
     \]

   - The Laplace transform of \( \cos(bt) \) is:
     \[
     \mathcal{L}\{\cos(bt)\} = \frac{s}{s^2 + b^2}
     \]
     For \( \frac{\cos(4t)}{2} \), we use \( b = 4 \), so:
     \[
     \mathcal{L}\left\{\frac{\cos(4t)}{2}\right\} = \frac{1}{2} \cdot \frac{s}{s^2 + 16}
     \]

4. **Combine the Results**:
   Adding the two terms together, the total Laplace transform is:
   \[
   \mathcal{L}\{f(t)\} = \frac{1}{2s} + \frac{s}{2(s^2 + 16)}
   \]

### Final Answer:
\[
\mathcal{L}\{ \cos^2(2t) \} = \frac{1}{2s} + \frac{s}{2(s^2 + 16)}
\]

Would you like more details or further clarifications? 

Here are some related questions for deeper understanding:
1. How does the Laplace transform handle functions involving higher powers of trigonometric functions?
2. What is the significance of the identity \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) in simplifying such problems?
3. How would you apply Laplace transform properties to other trigonometric identities, like \( \sin^2(t) \)?
4. What is the inverse Laplace transform of \( \frac{s}{2(s^2 + 16)} \)?
5. How does the Laplace transform of a sum of functions relate to the individual transforms?

**Tip**: When dealing with trigonometric functions in Laplace transforms, it's helpful to memorize common identities and transforms to simplify the process.

Es sobre las transformadas de Laplace, hay que resolverlo aplicando propiedades para f(t) = cos^2(2t).

This detailed solution explains how to find the Laplace transform of f(t) = cos^2(2t) by applying trigonometric identities and properties of the Laplace transform. Learn the step-by-step process, including simplifying the function and breaking it into terms.

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