Tacheometry is a method of surveying that is used to determine horizontal distances, vertical heights, and angular measurements. The tacheometric formula is a mathematical equation that relates the observed angles and the instrument constants to calculate the horizontal distances.

The tacheometric formula for horizontal distances using horizontal sights can be expressed as follows:

Horizontal Distance = Multiplying Constant * cos^2(α) + Additive Constant * cos(α)

Where:
- Multiplying Constant: This constant is multiplied by the square of the cosine of the observed angle (α).
- Additive Constant: This constant is multiplied by the cosine of the observed angle (α).

Explanation of the correct answer (Option D):

The correct answer is option D, which states that the multiplying constant is multiplied by cos^2(α), and the additive constant is multiplied by cos(α). This means that both the square of the cosine and the cosine of the observed angle are used in the tacheometric formula for horizontal distances.

The reason for using cos^2(α) is that it helps to account for the variation in the observed angle. The square of the cosine function provides a better representation of the angle than the cosine function alone, as it amplifies the effect of the angle on the horizontal distance.

On the other hand, the reason for using cos(α) is to introduce an additive constant that adjusts the horizontal distance based on the observed angle. The cosine function provides a scaling factor that accounts for the change in the horizontal distance due to the angle.

By combining the multiplying constant with cos^2(α) and the additive constant with cos(α), the tacheometric formula can accurately calculate the horizontal distances using horizontal sights.

In summary, the tacheometric formula for horizontal distances using horizontal sights involves using the multiplying constant with cos^2(α) and the additive constant with cos(α) to account for the observed angles and calculate the horizontal distances accurately.