To solve this problem, we can use the concept of similar triangles and trigonometry. Let's break down the problem into steps:

Step 1: Understanding the initial situation
- When the angle of elevation of the sun is 45 degrees, the length of the shadow of the person is x.
- We can assume that the person is standing upright, so the shadow is directly beneath the person.

Step 2: Finding the length of the person's height
- Since we have a right triangle formed by the person, the length of the shadow, and the angle of elevation, we can use trigonometry to find the length of the person's height.
- In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
- In this case, the height of the person is the opposite side, and the length of the shadow is the adjacent side.
- So, the tangent of 45 degrees is equal to the height of the person divided by the length of the shadow.
- Mathematically, this can be written as tan(45) = height/x.
- Since the tangent of 45 degrees is equal to 1, we have 1 = height/x.
- Therefore, the height of the person is also x.

Step 3: Understanding the change in the length of the shadow
- According to the problem, the length of the shadow increases by (√3-1) x.
- This means that the new length of the shadow is x + (√3-1) x = (1 + √3 - 1) x = √3 x.

Step 4: Finding the new angle of elevation
- We want to find the new angle of elevation of the sun when the length of the shadow is √3 x.
- Again, we can use trigonometry to solve this.
- In the new right triangle formed by the person, the length of the shadow, and the new angle of elevation, the tangent of the angle is equal to the height divided by the length of the shadow.
- Mathematically, this can be written as tan(new angle) = height/(√3 x).
- Since we know the height of the person is x (from Step 2), we can substitute this into the equation to get tan(new angle) = x/(√3 x).
- Simplifying the equation, we have tan(new angle) = 1/√3.
- Taking the inverse tangent of both sides of the equation, we find that the new angle is approximately 30 degrees.

Step 5: Determining the correct answer
- The correct answer is option D, which states that the new angle of elevation of the sun should be 30 degrees.
- This is consistent with the calculations we made in Step 4.

In conclusion, by using the concepts of similar triangles and trigonometry, we can determine that the angle of elevation of the sun should become 30 degrees when the length of the shadow increases by (√3-1) x.