The value of `"tan"1^(@)."tan"2^(@)."tan"3^(@)......."tan"89^(@)` is

Options

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

Answer

To find the value of \( \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ \), we can use the property of the tangent function.

### Step-by-Step Solution:

1. **Identify the Product**:
We need to evaluate the product \( P = \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ \).

2. **Pairing the Terms**:
Notice that \( \tan(90^\circ - \theta) = \cot \theta \). This means we can pair the terms:
- \( \tan 1^\circ \) pairs with \( \tan 89^\circ \)
- \( \tan 2^\circ \) pairs with \( \tan 88^\circ \)
- \( \tan 3^\circ \) pairs with \( \tan 87^\circ \)
- ...
- \( \tan 44^\circ \) pairs with \( \tan 46^\circ \)
- \( \tan 45^\circ \) stands alone.

3. **Using the Pairing Property**:
Each pair can be expressed as:
\[ \tan k^\circ \cdot \tan(90^\circ - k^\circ) = \tan k^\circ \cdot \cot k^\circ = 1 \]
Therefore, we can write:
\[ P = (\tan 1^\circ \cdot \tan 89^\circ) \cdot (\tan 2^\circ \cdot \tan 88^\circ) \cdots (\tan 44^\circ \cdot \tan 46^\circ) \cdot \tan 45^\circ \]

4. **Calculating the Product**:
Since each of the pairs evaluates to 1, we have:
\[ P = 1 \cdot 1 \cdots 1 \cdot \tan 45^\circ \]
And since \( \tan 45^\circ = 1 \):
\[ P = 1 \cdot 1 = 1 \]

5. **Final Result**:
Thus, the value of \( \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ \) is:
\[ \boxed{1} \]