The value of `tan1^@tan2^@tan3^@...tan89^@` is

Options

A. `0`
B. `1`
C. `(1)/(2)`
D. Not defined

Explanation

To find the value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ \), we can use the properties of the tangent function and its relationship with the cotangent function.

### Step-by-step Solution:

1. **Understanding the Product**:
We need to evaluate the product:
\[ P = \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ \]

2. **Pairing the Terms**:
Notice that \( \tan(90^\circ - x) = \cot x \). This means we can pair the terms:
\[ \tan 1^\circ \text{ with } \tan 89^\circ, \quad \tan 2^\circ \text{ with } \tan 88^\circ, \quad \ldots, \quad \tan 44^\circ \text{ with } \tan 46^\circ \]
The middle term is \( \tan 45^\circ \).

3. **Writing the Pairs**:
We can express the product as:
\[ P = (\tan 1^\circ \tan 89^\circ)(\tan 2^\circ \tan 88^\circ)(\tan 3^\circ \tan 87^\circ) \ldots (\tan 44^\circ \tan 46^\circ) \cdot \tan 45^\circ \]

4. **Using the Identity**:
For each pair, we have:
\[ \tan x \tan(90^\circ - x) = \tan x \cot x = 1 \]
Therefore, each pair contributes a value of 1 to the product.

5. **Counting the Pairs**:
There are 44 pairs (from \( \tan 1^\circ \) to \( \tan 44^\circ \)), and the middle term \( \tan 45^\circ = 1 \).

6. **Calculating the Final Product**:
Thus, the entire product can be simplified as:
\[ P = 1 \cdot 1 \cdot 1 \cdots \text{ (44 times)} \cdot 1 = 1 \]

### Conclusion:
The value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ \) is:
\[ \boxed{1} \]