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

Options

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

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 the symmetry of angles.

### Step-by-Step Solution:

1. **Understanding the Tangent Function:**
The tangent function has a property that \( \tan(90^\circ - x) = \cot x \). This means that for every angle \( x \), there is a complementary angle \( 90^\circ - x \) such that their tangents are reciprocals.

2. **Pairing the Angles:**
We can pair the angles 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 angle \( \tan 45^\circ \) is left unpaired.

3. **Calculating Each Pair:**
Each pair can be simplified using the property mentioned:
\[ \tan k^\circ \tan (90^\circ - k^\circ) = \tan k^\circ \cot k^\circ = 1 \]
Therefore, each of the pairs \( \tan k^\circ \tan (90^\circ - k^\circ) \) equals 1.

4. **Counting the Pairs:**
There are 44 pairs (from \( k = 1 \) to \( k = 44 \)), and we also have \( \tan 45^\circ \) in the middle, which equals 1.

5. **Final Calculation:**
The total product can be calculated as:
\[ \tan 1^\circ \tan 89^\circ \times \tan 2^\circ \tan 88^\circ \times \ldots \times \tan 44^\circ \tan 46^\circ \times \tan 45^\circ = 1^{44} \times 1 = 1 \]

Thus, the value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 89^\circ \) is \( \boxed{1} \).