What is the value of the following?
`tan 31^(@) tan 33^(@) tan 35^(@) …..tan57^(@) tan 59^(@)`

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To find the value of the expression \( \tan 31^\circ \tan 33^\circ \tan 35^\circ \ldots \tan 57^\circ \tan 59^\circ \), we can use the properties of tangent and cotangent functions.

### Step-by-Step Solution:

1. **Identify the Angles**:
The angles in the expression are \( 31^\circ, 33^\circ, 35^\circ, 37^\circ, 39^\circ, 41^\circ, 43^\circ, 45^\circ, 47^\circ, 49^\circ, 51^\circ, 53^\circ, 55^\circ, 57^\circ, 59^\circ \).

2. **Pair the Angles**:
Notice that \( \tan(90^\circ - x) = \cot(x) \). This means we can pair the angles such that:
- \( \tan 31^\circ \) pairs with \( \tan 59^\circ \)
- \( \tan 33^\circ \) pairs with \( \tan 57^\circ \)
- \( \tan 35^\circ \) pairs with \( \tan 55^\circ \)
- \( \tan 37^\circ \) pairs with \( \tan 53^\circ \)
- \( \tan 39^\circ \) pairs with \( \tan 51^\circ \)
- \( \tan 41^\circ \) pairs with \( \tan 49^\circ \)
- \( \tan 43^\circ \) pairs with \( \tan 47^\circ \)

3. **Evaluate Each Pair**:
Each pair can be evaluated as follows:
\[ \tan x \cdot \tan(90^\circ - x) = \tan x \cdot \cot x = 1 \]
Therefore, each of the pairs contributes a value of 1.

4. **Count the Pairs**:
There are 7 pairs (from \( 31^\circ \) to \( 59^\circ \)), and the middle angle \( \tan 45^\circ \) is equal to 1.

5. **Combine the Results**:
Since each of the 7 pairs contributes a value of 1 and the middle angle \( \tan 45^\circ = 1 \), the total product is:
\[ 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 = 1 \]

### Final Answer:
Thus, the value of \( \tan 31^\circ \tan 33^\circ \tan 35^\circ \ldots \tan 57^\circ \tan 59^\circ \) is \( \boxed{1} \).