Proof:

Let's consider a right triangle ABC, where AB represents the height of the tower, BC represents the distance from the observer to the tower, and AC represents the distance from the observer to the point one meter towards the tower.

The Angle of Elevation from the Ground:
The angle of elevation of the top of the tower as observed from a point on the ground is α. Let's consider the right triangle ABD, where AD represents the height of the tower and BD represents the distance from the observer to the tower.

In triangle ABD, we have:
tan(α) = AD/BD

The Angle of Elevation One Meter Towards the Tower:
The angle of elevation of the top of the tower as observed from a point one meter towards the tower is β. Let's consider the right triangle ACD, where AD represents the height of the tower and CD represents the distance from the observer to the point one meter towards the tower.

In triangle ACD, we have:
tan(β) = AD/CD

Deriving the Formula:
To find the height of the tower, we need to eliminate AD from the above two equations.

Multiplying the equation for tan(α) by tan(β), we get:
tan(α) * tan(β) = (AD/BD) * (AD/CD)
tan(α) * tan(β) = AD² / (BD * CD)

Rearranging the equation, we have:
AD² = (tan(α) * tan(β)) * (BD * CD)

Now, let's express BD as (CD + 1) since BC is the distance from the observer to the tower, and CD is the distance from the observer to the point one meter towards the tower.
BD = CD + 1

Substituting BD in terms of CD in the equation, we get:
AD² = (tan(α) * tan(β)) * (CD * (CD + 1))

Final Calculation:
To find the height of the tower, we need to find AD. Taking the square root of both sides of the equation, we have:
AD = √[(tan(α) * tan(β)) * (CD * (CD + 1))]

Therefore, the height of the tower is given by:
Height = AD = √[(tan(α) * tan(β)) * (CD * (CD + 1))]

Simplifying further, we get:
Height = (tan(α) * tan(β)) / √[(1 / (CD * (CD + 1)))]

Since tan(β) = AD/CD, we can rewrite the equation as:
Height = (tan(α) * tan(β)) / √[(1 / (AD/CD * (AD/CD + 1)))]

Simplifying the denominator, we have:
Height = (tan(α) * tan(β)) / √[(1 / (AD²/CD² + AD/CD))]

Since AD² = (tan(α) * tan(β)) * (BD * CD), we can rewrite the equation as:
Height = (tan(α) * tan(β)) / √[(1 / (tan(α) * tan(β) * (BD * CD