If `(1+tan1^(@)) *(1+tan2^(@))*(1+tan3^(@)) …….(1+tan45^(@)) = 2^(n)`, then 'n' is equal to :

Options

A. 16
B. 23
C. 30
D. None of these

Answer

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

\[ (1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \ldots (1 + \tan 45^\circ) = 2^n \]

### Step-by-Step Solution:

1. **Understanding the Identity**:
We will use the identity:
\[ (1 + \tan x)(1 + \tan(45^\circ - x)) = 2 \]
This identity holds because:
\[ \tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x} \]
Thus,
\[ (1 + \tan x)(1 + \tan(45^\circ - x)) = (1 + \tan x)\left(1 + \frac{1 - \tan x}{1 + \tan x}\right) = 2 \]

2. **Pairing Terms**:
We can pair the terms in the product:
- \( (1 + \tan 1^\circ)(1 + \tan 44^\circ) \)
- \( (1 + \tan 2^\circ)(1 + \tan 43^\circ) \)
- \( (1 + \tan 3^\circ)(1 + \tan 42^\circ) \)
- ...
- \( (1 + \tan 22^\circ)(1 + \tan 23^\circ) \)
- The last term is \( (1 + \tan 45^\circ) \)

Each of these pairs evaluates to 2, and there are 22 pairs.

3. **Calculating the Product**:
Since there are 22 pairs, we have:
\[ 2^{22} \text{ from the pairs} \]
Additionally, we have the last term:
\[ 1 + \tan 45^\circ = 1 + 1 = 2 \]

4. **Combining the Results**:
Therefore, the entire product can be expressed as:
\[ 2^{22} \times 2^1 = 2^{22 + 1} = 2^{23} \]

5. **Finding 'n'**:
From the equation \( 2^{23} = 2^n \), we can conclude that:
\[ n = 23 \]

### Final Answer:
The value of \( n \) is \( \boxed{23} \).