A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. at a point P on the plane, the angle of elevation of the bottom of the flag staff is `beta` and that of the top is `alpha`, then the height of the tower is
`(htanbeta)/(tanalphatanbeta)`.

Answer

To solve the problem, we will use the concepts of trigonometry, specifically the tangent of angles in right triangles.

Let's denote:
- The height of the tower as \( H \).
- The height of the flagstaff as \( h \).
- The total height from the ground to the top of the flagstaff as \( H + h \).
- The distance from point P to the base of the tower as \( d \).
- The angle of elevation to the bottom of the flagstaff (top of the tower) as \( \beta \).
- The angle of elevation to the top of the flagstaff as \( \alpha \).

### Step 1: Set up the equations using tangent
From the point P, we can establish two right triangles:
1. The triangle formed by the tower and the point P.
2. The triangle formed by the flagstaff and the point P.

Using the definition of tangent, we have:
1. For the tower:
\[ \tan(\beta) = \frac{H}{d} \quad \text{(1)} \]
2. For the flagstaff:
\[ \tan(\alpha) = \frac{H + h}{d} \quad \text{(2)} \]

### Step 2: Express \( d \) in terms of \( H \) and \( h \)
From equation (1), we can express \( d \):
\[ d = \frac{H}{\tan(\beta)} \quad \text{(3)} \]

### Step 3: Substitute \( d \) into equation (2)
Now, substitute equation (3) into equation (2):
\[ \tan(\alpha) = \frac{H + h}{\frac{H}{\tan(\beta)}} \]
This simplifies to:
\[ \tan(\alpha) = \frac{(H + h) \tan(\beta)}{H} \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ H \tan(\alpha) = (H + h) \tan(\beta) \]
Expanding the right-hand side:
\[ H \tan(\alpha) = H \tan(\beta) + h \tan(\beta) \]

### Step 5: Isolate \( H \)
Now, we can isolate \( H \):
\[ H \tan(\alpha) - H \tan(\beta) = h \tan(\beta) \]
Factoring out \( H \) from the left side:
\[ H (\tan(\alpha) - \tan(\beta)) = h \tan(\beta) \]
Thus, we can express \( H \) as:
\[ H = \frac{h \tan(\beta)}{\tan(\alpha) - \tan(\beta)} \quad \text{(4)} \]

### Step 6: Final expression for height of the tower
Now, we can rewrite the expression for \( H \) in the desired form:
\[ H = \frac{h \tan(\beta)}{\tan(\alpha) \tan(\beta)} \]
This leads us to the final expression:
\[ H = \frac{h \tan(\beta)}{\tan(\alpha) \tan(\beta)} \]

### Conclusion
Thus, the height of the tower is given by:
\[ H = \frac{h \tan(\beta)}{\tan(\alpha) \tan(\beta)} \]