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

Options

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

Explanation

To find the value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ \), we can follow these steps:

### Step-by-step Solution:

1. **Group the Terms**:
We can group the terms in pairs such that each pair adds up to \(90^\circ\):
\[ (\tan 1^\circ \times \tan 89^\circ), (\tan 2^\circ \times \tan 88^\circ), \ldots, (\tan 44^\circ \times \tan 46^\circ) \]
The middle term, \( \tan 45^\circ \), will be left alone.

2. **Use the Identity**:
Recall the trigonometric identity:
\[ \tan(90^\circ - \theta) = \cot \theta \]
Therefore, we can rewrite each pair:
\[ \tan 89^\circ = \cot 1^\circ, \quad \tan 88^\circ = \cot 2^\circ, \quad \tan 87^\circ = \cot 3^\circ, \ldots, \quad \tan 46^\circ = \cot 44^\circ \]

3. **Rewrite the Product**:
Using the identity, we can express the product as:
\[ \tan 1^\circ \times \cot 1^\circ, \quad \tan 2^\circ \times \cot 2^\circ, \quad \ldots, \quad \tan 44^\circ \times \cot 44^\circ \]
This gives us:
\[ (\tan 1^\circ \times \cot 1^\circ) \times (\tan 2^\circ \times \cot 2^\circ) \times \ldots \times (\tan 44^\circ \times \cot 44^\circ) \times \tan 45^\circ \]

4. **Evaluate Each Pair**:
Each pair \( \tan \theta \times \cot \theta = 1 \):
\[ \tan 1^\circ \times \cot 1^\circ = 1, \quad \tan 2^\circ \times \cot 2^\circ = 1, \quad \ldots, \quad \tan 44^\circ \times \cot 44^\circ = 1 \]
Thus, we have \( 44 \) pairs contributing \( 1 \) each.

5. **Include \( \tan 45^\circ \)**:
We know that:
\[ \tan 45^\circ = 1 \]

6. **Final Calculation**:
Therefore, the entire product simplifies to:
\[ 1 \times 1 \times 1 \times \ldots \times 1 \times 1 = 1 \]
(where there are \( 44 \) ones from the pairs and one from \( \tan 45^\circ \)).

### Final Answer:
The value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ = 1 \).

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