If `e^(( cos^2 x + cos^4x+ cos ^6 x + ……..oo ) log_e 2)` satisfies the equation ` t^2 - 9t + 8=0` , then the value of `( 2 sin x)/( sin x + sqrt(3) cos x) ( 0 lt x lt (pi)/(2))` is

Options

A. `2 sqrt(3)`
B. `3/2`
C. `sqrt(3)`
D. `1/2`

Answer

To solve the given problem step by step, let's break down the components of the question.

### Step 1: Analyze the Infinite Series
The expression given is:
\[ e^{(\cos^2 x + \cos^4 x + \cos^6 x + \ldots) \log_e 2} \]

This is an infinite geometric series where:
- The first term \( a = \cos^2 x \)
- The common ratio \( r = \cos^2 x \)

The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} = \frac{\cos^2 x}{1 - \cos^2 x} = \frac{\cos^2 x}{\sin^2 x} = \cot^2 x \]

Thus, we can rewrite the expression as:
\[ e^{\cot^2 x \log_e 2} = 2^{\cot^2 x} \]

### Step 2: Set Up the Equation
From the problem, we know that this expression satisfies the equation:
\[ t^2 - 9t + 8 = 0 \]

To find the roots of this quadratic equation, we can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -9, c = 8 \).

Calculating the discriminant:
\[ b^2 - 4ac = 81 - 32 = 49 \]
Calculating the roots:
\[ t = \frac{9 \pm 7}{2} = \{8, 1\} \]

### Step 3: Relate Roots to the Expression
The expression \( 2^{\cot^2 x} \) must equal one of the roots:
1. If \( 2^{\cot^2 x} = 8 \):
\[ \cot^2 x = 3 \implies \cot x = \sqrt{3} \implies x = \frac{\pi}{6} \]
2. If \( 2^{\cot^2 x} = 1 \):
\[ \cot^2 x = 0 \implies \cot x = 0 \implies x = \frac{\pi}{2} \]

Since \( 0 < x < \frac{\pi}{2} \), we take \( x = \frac{\pi}{6} \).

### Step 4: Calculate the Required Expression
Now we need to find the value of:
\[ \frac{2 \sin x}{\sin x + \sqrt{3} \cos x} \]
Substituting \( x = \frac{\pi}{6} \):
\[ \sin \frac{\pi}{6} = \frac{1}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \]
Thus, the expression becomes:
\[ \frac{2 \cdot \frac{1}{2}}{\frac{1}{2} + \sqrt{3} \cdot \frac{\sqrt{3}}{2}} = \frac{1}{\frac{1}{2} + \frac{3}{2}} = \frac{1}{2} \]

### Final Answer
The value of the expression is:
\[ \frac{1}{2} \]