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We are given that:

\[
\tan(\alpha) = \frac{12}{5}
\]

We need to find the approximate value of the expression:

\[
13 \cdot \cos(\alpha) + \cot(\alpha)
\]

### Step 1: Use the given information to find \(\cos(\alpha)\) and \(\cot(\alpha)\)

We know that:

\[
\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{12}{5}
\]

From the tangent identity, we can set:

\[
\sin(\alpha) = 12k \quad \text{and} \quad \cos(\alpha) = 5k
\]

Now, using the Pythagorean identity:

\[
\sin^2(\alpha) + \cos^2(\alpha) = 1
\]

Substitute \(\sin(\alpha) = 12k\) and \(\cos(\alpha) = 5k\):

\[
(12k)^2 + (5k)^2 = 1
\]

\[
144k^2 + 25k^2 = 1
\]

\[
169k^2 = 1 \quad \Rightarrow \quad k^2 = \frac{1}{169} \quad \Rightarrow \quad k = \frac{1}{13}
\]

Thus:

\[
\sin(\alpha) = \frac{12}{13}, \quad \cos(\alpha) = \frac{5}{13}
\]

### Step 2: Find \(\cot(\alpha)\)

\(\cot(\alpha)\) is the reciprocal of \(\tan(\alpha)\):

\[
\cot(\alpha) = \frac{1}{\tan(\alpha)} = \frac{5}{12}
\]

### Step 3: Calculate the expression

Now, substitute the values of \(\cos(\alpha)\) and \(\cot(\alpha)\) into the given expression:

\[
13 \cdot \cos(\alpha) + \cot(\alpha) = 13 \cdot \frac{5}{13} + \frac{5}{12}
\]

\[
= 5 + \frac{5}{12} = 5 + 0.4167 \approx 5.4167
\]

### Conclusion:

The approximate value of the expression is \(5.42\), so the correct answer is **A**.

---

Would you like further details on any step? Here are five related questions for further exploration:

1. How is the Pythagorean identity used to find trigonometric functions?
2. What are other ways to solve for \(\cos(\alpha)\) and \(\cot(\alpha)\)?
3. How can we convert between different trigonometric functions?
4. Why do we set \(\sin(\alpha)\) and \(\cos(\alpha)\) in terms of a constant \(k\)?
5. What are the applications of trigonometric identities in solving real-world problems?

**Tip:** Always double-check your results when dealing with approximations in trigonometry to ensure accuracy.

Sabiendo que tan(α) = 12/5, entonces el valor aproximado de la expresión 13 · cos(α) + cot(α) = 

Solve the trigonometric expression where tan(α) = 12/5. Learn how to find the value of 13 · cos(α) + cot(α) using Pythagorean identity and trigonometric formulas. Suitable for Grades 10-12.

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