If `x in(1, 30)` and `(1+tan x^(@))(1+tan(x+1)^(@))...(1+tan(x+44)^(@)) = 2^(23)` then value of x is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To solve the problem, we need to evaluate the expression given in the question step by step.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We need to evaluate the expression:
\[ (1 + \tan x^\circ)(1 + \tan(x + 1)^\circ)(1 + \tan(x + 2)^\circ) \ldots (1 + \tan(x + 44)^\circ) = 2^{23} \]

2. **Setting Up the Function**:
Let's define:
\[ f(x) = (1 + \tan x^\circ)(1 + \tan(x + 1)^\circ)(1 + \tan(x + 2)^\circ) \ldots (1 + \tan(x + 44)^\circ) \]

3. **Substituting Values**:
We will first evaluate \( f(1) \):
\[ f(1) = (1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \ldots (1 + \tan 45^\circ) \]

4. **Using the Identity**:
Notice that:
\[ \tan(45^\circ - a) = \frac{1 - \tan a}{1 + \tan a} \]
This means:
\[ 1 + \tan(45^\circ - a) = \frac{2}{1 + \tan a} \]
Therefore, we can pair terms:
\[ (1 + \tan 1^\circ)(1 + \tan 44^\circ), (1 + \tan 2^\circ)(1 + \tan 43^\circ), \ldots, (1 + \tan 22^\circ)(1 + \tan 23^\circ) \]

5. **Calculating the Pairs**:
Each pair can be simplified:
\[ (1 + \tan a)(1 + \tan(45^\circ - a)) = (1 + \tan a)\left( \frac{2}{1 + \tan a} \right) = 2 \]
There are 22 such pairs, and one additional term \( (1 + \tan 45^\circ) = 2 \).

6. **Final Calculation**:
Thus, we have:
\[ f(1) = 2^{22} \cdot 2 = 2^{23} \]

7. **Conclusion**:
Since \( f(1) = 2^{23} \), we conclude that the value of \( x \) that satisfies the equation is:
\[ x = 1 \]

### Final Answer:
The value of \( x \) is \( \boxed{1} \).