Aclock tower stands at the crossing of two roads whichpoint in the north-south and the east-westdirections. P, Q, R and S are points on the roads due north, east, south and west respectively, where the angles of elevation of the top of the tower are respectively,
`alpha, beta, gamma and delta, "Then " ((PQ)/(RS))^2` is equal to :

Options

A. `{:(tan^2alpha + tan^2 delta)/(tan^2beta +tan^2 gamma):}`
B. `{:(tan^2 alpha + tan^2beta)/(tan^2 gamma+ tan^2 delta):}`
C. `{:(cot^2alpha + cot^2 beta)/(cot^2 gamma + cot^2 delta):}`
D. `{:(cot^2 alpha + cot^2 delta)/(cot^2 beta + cot^2 gamma):}`

Answer

To solve the problem, we need to analyze the situation involving the clock tower and the angles of elevation from points P, Q, R, and S. Let's break it down step by step.

### Step 1: Understand the Geometry
We have a clock tower standing at the intersection of two roads, with points P, Q, R, and S located to the north, east, south, and west of the tower, respectively. The angles of elevation from these points to the top of the tower are given as α (alpha), β (beta), γ (gamma), and δ (delta).

### Step 2: Define Variables
Let:
- H = height of the clock tower
- OP = distance from the tower to point P
- OQ = distance from the tower to point Q
- OR = distance from the tower to point R
- OS = distance from the tower to point S

### Step 3: Use Trigonometric Ratios
From the definition of tangent in right triangles, we can express the distances in terms of the height of the tower and the angles of elevation:
- From point P:
\[ \tan(\alpha) = \frac{H}{OP} \implies OP = \frac{H}{\tan(\alpha)} = H \cot(\alpha) \]

- From point Q:
\[ \tan(\beta) = \frac{H}{OQ} \implies OQ = \frac{H}{\tan(\beta)} = H \cot(\beta) \]

- From point R:
\[ \tan(\gamma) = \frac{H}{OR} \implies OR = \frac{H}{\tan(\gamma)} = H \cot(\gamma) \]

- From point S:
\[ \tan(\delta) = \frac{H}{OS} \implies OS = \frac{H}{\tan(\delta)} = H \cot(\delta) \]

### Step 4: Calculate Distances PQ and RS
Using the Pythagorean theorem:
- For PQ:
\[ PQ^2 = OP^2 + OQ^2 = (H \cot(\alpha))^2 + (H \cot(\beta))^2 = H^2 (\cot^2(\alpha) + \cot^2(\beta)) \]

- For RS:
\[ RS^2 = OS^2 + OR^2 = (H \cot(\delta))^2 + (H \cot(\gamma))^2 = H^2 (\cot^2(\delta) + \cot^2(\gamma)) \]

### Step 5: Find the Ratio \((PQ/RS)^2\)
Now we can find the ratio of the squares:
\[ \left(\frac{PQ}{RS}\right)^2 = \frac{PQ^2}{RS^2} = \frac{H^2 (\cot^2(\alpha) + \cot^2(\beta))}{H^2 (\cot^2(\delta) + \cot^2(\gamma))} \]
The \(H^2\) cancels out:
\[ \left(\frac{PQ}{RS}\right)^2 = \frac{\cot^2(\alpha) + \cot^2(\beta)}{\cot^2(\delta) + \cot^2(\gamma)} \]

### Final Answer
Thus, the final result is:
\[ \left(\frac{PQ}{RS}\right)^2 = \frac{\cot^2(\alpha) + \cot^2(\beta)}{\cot^2(\delta) + \cot^2(\gamma)} \]