If `theta = 9^(@),` then what is the value of
`cot theta cot 2 theta cot 3 theta cot 4 theta cot 5 theta cot 6 theta cot 7 theta cot 8 theta cot 9 theta` ?

Options

A. `(1)/(sqrt3)`
B. `sqrt3 -1`
C. 1
D. `sqrt3`

Answer

To solve the problem, we need to find the value of the product:

\[ \cot \theta \cdot \cot 2\theta \cdot \cot 3\theta \cdot \cot 4\theta \cdot \cot 5\theta \cdot \cot 6\theta \cdot \cot 7\theta \cdot \cot 8\theta \cdot \cot 9\theta \]

where \(\theta = 9^\circ\).

### Step 1: Identify the angles
We can express the angles in terms of \(\theta\):
- \(\cot \theta = \cot 9^\circ\)
- \(\cot 2\theta = \cot 18^\circ\)
- \(\cot 3\theta = \cot 27^\circ\)
- \(\cot 4\theta = \cot 36^\circ\)
- \(\cot 5\theta = \cot 45^\circ\)
- \(\cot 6\theta = \cot 54^\circ\)
- \(\cot 7\theta = \cot 63^\circ\)
- \(\cot 8\theta = \cot 72^\circ\)
- \(\cot 9\theta = \cot 81^\circ\)

### Step 2: Use the cotangent identity
We know that:
\[ \cot A \cdot \cot (90^\circ - A) = 1 \]
This means:
- \(\cot 9^\circ \cdot \cot 81^\circ = 1\)
- \(\cot 18^\circ \cdot \cot 72^\circ = 1\)
- \(\cot 27^\circ \cdot \cot 63^\circ = 1\)
- \(\cot 36^\circ \cdot \cot 54^\circ = 1\)

### Step 3: Group the cotangents
Now, we can group the cotangents:
\[ (\cot 9^\circ \cdot \cot 81^\circ) \cdot (\cot 18^\circ \cdot \cot 72^\circ) \cdot (\cot 27^\circ \cdot \cot 63^\circ) \cdot (\cot 36^\circ \cdot \cot 54^\circ) \cdot \cot 45^\circ \]

### Step 4: Calculate the product
Each of the pairs of cotangents equals 1:
\[ 1 \cdot 1 \cdot 1 \cdot 1 \cdot \cot 45^\circ \]
Since \(\cot 45^\circ = 1\), we have:
\[ 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]