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To prove that \(\cos \alpha = \frac{5}{13}\), let's analyze the given diagram.

The triangle \(PMN\) is a right triangle with \(M\) as the vertex of the right angle, so \(\angle M\) is \(90^\circ\).

### Given:
- \(PM = 15\)
- \(MN = 13\)
- \(PN = 9 + 5 = 14\) (since \(OP = 9\) and \(ON = 5\), we will need to verify if the sum of these is consistent with what is given in the problem)

However, there's no direct need to sum these, and it appears there's a focus on segment \(ON\), since \(\alpha\) is the angle between \(OM\) and \(ON\).

### Steps to prove:

1. **Identify the right triangle relevant to angle \(\alpha\):**

   Consider \(\triangle MON\), where:
   - \(MO = 13\) (hypotenuse)
   - \(ON = 5\) (adjacent side to angle \(\alpha\))
  
2. **Apply the definition of cosine:**

   The cosine of angle \(\alpha\) is given by the ratio of the length of the adjacent side to the hypotenuse:
   \[
   \cos \alpha = \frac{\text{Adjacent side (ON)}}{\text{Hypotenuse (MO)}} = \frac{ON}{MO} = \frac{5}{13}
   \]

Thus, \(\cos \alpha = \frac{5}{13}\), as required.

Would you like more details on this proof or have any questions?

### Related Questions:
1. How can we verify the side lengths of a right triangle?
2. What other trigonometric functions can be calculated for angle \(\alpha\) in this triangle?
3. How is the sine of angle \(\alpha\) determined in this context?
4. What is the length of side \(OM\) using the Pythagorean theorem?
5. Can we determine the area of triangle \(PMN\) using trigonometric identities?

### Tip:
When working with trigonometric identities, always make sure to correctly identify the sides relative to the angle of interest.

Learn how to prove that \(\cos \alpha = \frac{5}{13}\) using trigonometric principles. Detailed step-by-step solution analyzing a right triangle and applying the cosine formula. Suitable for high school students studying trigonometry.

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