Find the value of tan `1^@` tan `2^@` tan `3^@` ….tan `45^@` …..tan `88^@` tan `89^@`

Answer

To find the value of the product \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 45^\circ \ldots \tan 88^\circ \tan 89^\circ \), we can use the property of the tangent function and its relationship with cotangent.

### Step-by-Step Solution:

1. **Understanding the Product**:
The product can be expressed as:
\[ P = \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 45^\circ \ldots \tan 88^\circ \tan 89^\circ \]

2. **Pairing Terms**:
Notice that \( \tan(90^\circ - x) = \cot x \). Therefore, we can pair the terms:
\[ \tan 1^\circ \tan 89^\circ, \tan 2^\circ \tan 88^\circ, \tan 3^\circ \tan 87^\circ, \ldots, \tan 44^\circ \tan 46^\circ \]
and we have \( \tan 45^\circ \) in the middle.

3. **Using the Cotangent Identity**:
Each pair can be rewritten using the cotangent identity:
\[ \tan x \tan(90^\circ - x) = \tan x \cot x = 1 \]
Thus, we can express the product as:
\[ P = (\tan 1^\circ \tan 89^\circ) \times (\tan 2^\circ \tan 88^\circ) \times \ldots \times (\tan 44^\circ \tan 46^\circ) \times \tan 45^\circ \]

4. **Calculating the Number of Pairs**:
There are 44 pairs from \( \tan 1^\circ \tan 89^\circ \) to \( \tan 44^\circ \tan 46^\circ \) and one middle term \( \tan 45^\circ \):
\[ P = 1 \times 1 \times \ldots \times 1 \times 1 = 1 \]
(since each pair contributes a product of 1).

5. **Final Calculation**:
Since \( \tan 45^\circ = 1 \):
\[ P = 1 \times 1 = 1 \]

### Conclusion:
The value of \( \tan 1^\circ \tan 2^\circ \tan 3^\circ \ldots \tan 45^\circ \ldots \tan 88^\circ \tan 89^\circ \) is:
\[ \boxed{1} \]