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 equation

\[ (1 + \tan(x^\circ))(1 + \tan((x+1)^\circ))(1 + \tan((x+2)^\circ)) \ldots (1 + \tan((x+44)^\circ)) = 2^{23} \]

where \(x \in (1, 30)\), we will evaluate the left-hand side by substituting values for \(x\).

### Step 1: Substitute \(x = 1\)

Let's first substitute \(x = 1\):

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

### Step 2: Simplify the expression

We know that \(\tan(45^\circ) = 1\), so:

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

Now, we can pair the terms in the product:

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

### Step 3: Use the identity

Using the identity:

\[ 1 + \tan(A) = \frac{1 + \tan(A)}{1 - \tan(A)} \text{ when } A + B = 45^\circ \]

we can simplify each pair:

\[ (1 + \tan(k^\circ))(1 + \tan((45 - k)^\circ)) = 2 \]

for \(k = 1, 2, \ldots, 22\).

### Step 4: Count the pairs

There are 22 pairs from \(1^\circ\) to \(22^\circ\) and one additional term \(1 + \tan(45^\circ)\):

\[ \text{Total product} = 2^{22} \cdot 2 = 2^{23} \]

### Step 5: Conclusion

Since we have shown that substituting \(x = 1\) satisfies the equation, we conclude that:

\[ \text{The value of } x \text{ is } 1. \]