The value of cot `15^(@) cot 16^(@) cot 17^(@) . . . . . cot 73^(@) cot 74^(@) cot 75^(@)` is :

Options

A. `(1)/(2)`
B. 0
C. 1
D. `-1`

Answer

To find the value of \( \cot 15^\circ \cot 16^\circ \cot 17^\circ \ldots \cot 73^\circ \cot 74^\circ \cot 75^\circ \), we can use the identity of cotangent:

\[ \cot(90^\circ - \theta) = \tan(\theta) \]

### Step-by-Step Solution:

1. **Identify the Range of Angles**:
We need to evaluate the product of cotangents from \( 15^\circ \) to \( 75^\circ \).

2. **Pair the Cotangent Values**:
We can pair the cotangent values using the identity mentioned above:
- \( \cot 15^\circ = \cot(90^\circ - 75^\circ) = \tan 75^\circ \)
- \( \cot 16^\circ = \cot(90^\circ - 74^\circ) = \tan 74^\circ \)
- \( \cot 17^\circ = \cot(90^\circ - 73^\circ) = \tan 73^\circ \)
- ...
- \( \cot 74^\circ = \cot(90^\circ - 16^\circ) = \tan 16^\circ \)
- \( \cot 75^\circ = \cot(90^\circ - 15^\circ) = \tan 15^\circ \)

3. **Rewrite the Product**:
The product can be rewritten as:
\[ \cot 15^\circ \cot 75^\circ \cdot \cot 16^\circ \cot 74^\circ \cdots \cot 45^\circ \]

4. **Evaluate Each Pair**:
Each pair \( \cot x^\circ \cot (90^\circ - x^\circ) = 1 \):
- For example, \( \cot 15^\circ \cot 75^\circ = 1 \)
- \( \cot 16^\circ \cot 74^\circ = 1 \)
- Continuing this way, we find that all pairs multiply to 1.

5. **Consider the Middle Value**:
The middle angle is \( \cot 45^\circ \), which is also equal to 1.

6. **Final Calculation**:
Since all pairs equal 1 and the middle value \( \cot 45^\circ = 1 \), the entire product simplifies to:
\[ 1 \times 1 \times 1 \ldots = 1 \]

Thus, the final value of \( \cot 15^\circ \cot 16^\circ \ldots \cot 75^\circ \) is:
\[ \boxed{1} \]