Value of `(tan 1^@ tan2^@ tan3^@...tan89^@)` is :

Options

A. 1
B. 89
C. undefined
D. 0

Answer

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 cotangent.

### Step-by-step Solution:

1. **Understanding the Symmetry**:
The tangent function has a property where \( \tan(90^\circ - x) = \cot x \). This means that for every angle \( x \) in the range from \( 1^\circ \) to \( 89^\circ \), there is a corresponding angle \( 90^\circ - x \) such that:
\[ \tan(90^\circ - x) = \cot x \]

2. **Pairing the Angles**:
We can pair the terms in the product:
\[ \tan 1^\circ \tan 89^\circ, \tan 2^\circ \tan 88^\circ, \tan 3^\circ \tan 87^\circ, \ldots, \tan 44^\circ \tan 46^\circ \]
The middle term \( \tan 45^\circ \) stands alone since \( \tan 45^\circ = 1 \).

3. **Using the Cotangent Identity**:
Each pair can be expressed using the cotangent identity:
\[ \tan k^\circ \tan(90^\circ - k^\circ) = \tan k^\circ \cot k^\circ = 1 \]
Therefore:
\[ \tan 1^\circ \tan 89^\circ = 1, \quad \tan 2^\circ \tan 88^\circ = 1, \quad \tan 3^\circ \tan 87^\circ = 1, \ldots, \tan 44^\circ \tan 46^\circ = 1 \]

4. **Counting the Pairs**:
There are \( 44 \) pairs (from \( 1^\circ \) to \( 44^\circ \)) and one \( \tan 45^\circ \):
\[ \text{Total product} = (\tan 1^\circ \tan 89^\circ) \times (\tan 2^\circ \tan 88^\circ) \times \ldots \times (\tan 44^\circ \tan 46^\circ) \times \tan 45^\circ \]
Since each pair equals \( 1 \) and \( \tan 45^\circ = 1 \), the total product is:
\[ 1 \times 1 \times \ldots \times 1 \times 1 = 1 \]

5. **Conclusion**:
Therefore, the value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ \) is:
\[ \boxed{1} \]