If the value of
`(1+tan 1^(@))(1+ tan 2^(@)) (1+tan 3^(@))`……….(1+ tan 45) Is `2^(lambda)`, then the sum of the digits of the number `lambda` is

Options

A. 3
B. 6
C. 5
D. 4

Answer

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

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

and find the value of \(\lambda\) such that this expression equals \(2^\lambda\).

### Step 1: Understanding the Identity
We can use the identity derived from the angle addition formula for tangent. If \(a + b = 45^\circ\), then:

\[ (1 + \tan a)(1 + \tan b) = 2 \]

This means that for any two angles \(a\) and \(b\) such that \(a + b = 45^\circ\), the product of \(1 + \tan a\) and \(1 + \tan b\) equals 2.

### Step 2: Pairing the Terms
We can pair the terms in our original product:

- Pair \(1^\circ\) with \(44^\circ\)
- Pair \(2^\circ\) with \(43^\circ\)
- Pair \(3^\circ\) with \(42^\circ\)
- ...
- Pair \(22^\circ\) with \(23^\circ\)

This gives us the following pairs:

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

### Step 3: Counting the Pairs
Each of these pairs evaluates to 2, and there are 22 pairs from \(1^\circ\) to \(22^\circ\). Additionally, we have the term \(1 + \tan 45^\circ\):

\[ 1 + \tan 45^\circ = 1 + 1 = 2 \]

### Step 4: Total Product Calculation
Thus, the total product can be expressed as:

\[ (1 + \tan 1^\circ)(1 + \tan 2^\circ) \ldots (1 + \tan 45^\circ) = 2^{22} \times 2 = 2^{23} \]

### Step 5: Finding \(\lambda\)
From the equation, we have:

\[ 2^{\lambda} = 2^{23} \]

Thus, \(\lambda = 23\).

### Step 6: Sum of the Digits of \(\lambda\)
Now, we need to find the sum of the digits of \(\lambda\):

\[ \text{Digits of } 23: 2, 3 \]

Sum of the digits:

\[ 2 + 3 = 5 \]

### Final Answer
The sum of the digits of the number \(\lambda\) is:

\[ \boxed{5} \]