The angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height is `45^(@)`.

Answer

To determine whether the statement "The angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height is 45 degrees" is true or false, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Problem**: We have a vertical pole of height \( h \) and a shadow of length \( h \). We need to find the angle of elevation of the sun.

2. **Draw a Diagram**:
- Let \( AB \) be the vertical pole with height \( h \).
- Let \( BC \) be the shadow on the ground, which also has length \( h \).
- The angle of elevation of the sun from point \( C \) (the tip of the shadow) to point \( A \) (the top of the pole) is denoted as \( \alpha \).

3. **Identify the Right Triangle**:
- The triangle \( ABC \) is a right triangle where:
- \( AB \) (the height of the pole) is opposite to angle \( \alpha \).
- \( BC \) (the length of the shadow) is adjacent to angle \( \alpha \).

4. **Use the Tangent Function**:
- The tangent of angle \( \alpha \) is given by the ratio of the opposite side to the adjacent side:
\[ \tan(\alpha) = \frac{AB}{BC} \]

5. **Substitute the Values**:
- Since \( AB = h \) and \( BC = h \), we have:
\[ \tan(\alpha) = \frac{h}{h} = 1 \]

6. **Find the Angle**:
- We know that \( \tan(45^\circ) = 1 \). Therefore, we can conclude:
\[ \alpha = 45^\circ \]

7. **Conclusion**:
- Since we have determined that the angle of elevation \( \alpha \) is indeed \( 45^\circ \), the statement is **true**.

### Final Answer:
The statement is **True**.