The value of `(1+ tan 1^(@)) (1+ tan 44^(@)) (1+ tan 2^(@)) (1+ tan 43^(@))…(1+ tan 22^(@)) (1+ tan 23^(@))` is

Options

A. 1
B. `2^(22)`
C. `3^(8)`
D. `2^(11)`

Answer

To solve the problem, we need to evaluate the expression:

\[ (1 + \tan 1^\circ)(1 + \tan 44^\circ)(1 + \tan 2^\circ)(1 + \tan 43^\circ) \cdots (1 + \tan 22^\circ)(1 + \tan 23^\circ) \]

### Step-by-Step Solution:

1. **Pair the Terms:**
Notice that the angles can be paired as follows:
- \( (1^\circ, 44^\circ) \)
- \( (2^\circ, 43^\circ) \)
- \( (3^\circ, 42^\circ) \)
- ...
- \( (22^\circ, 23^\circ) \)

Each pair sums up to \( 45^\circ \).

2. **Use the Identity:**
For each pair \( (x, 45^\circ - x) \), we can use the identity:
\[ \tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x} \]
Therefore, we can express \( 1 + \tan(45^\circ - x) \) as:
\[ 1 + \tan(45^\circ - x) = 1 + \frac{1 - \tan x}{1 + \tan x} = \frac{(1 + \tan x) + (1 - \tan x)}{1 + \tan x} = \frac{2}{1 + \tan x} \]

3. **Evaluate Each Pair:**
For the first pair \( (1^\circ, 44^\circ) \):
\[ (1 + \tan 1^\circ)(1 + \tan 44^\circ) = (1 + \tan 1^\circ) \left( \frac{2}{1 + \tan 1^\circ} \right) = 2 \]

Similarly, for the second pair \( (2^\circ, 43^\circ) \):
\[ (1 + \tan 2^\circ)(1 + \tan 43^\circ) = (1 + \tan 2^\circ) \left( \frac{2}{1 + \tan 2^\circ} \right) = 2 \]

Continuing this process for all pairs, we find that each pair evaluates to \( 2 \).

4. **Count the Pairs:**
There are \( 22 \) pairs in total (from \( 1^\circ \) to \( 22^\circ \) paired with \( 44^\circ \) to \( 23^\circ \)).

5. **Final Calculation:**
Since each of the \( 22 \) pairs contributes a factor of \( 2 \), the total product is:
\[ 2^{22} \]

### Conclusion:
Thus, the value of the expression is:

\[ \boxed{2^{22}} \]