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Q.

If $e^{(\cos^{2} x + \cos^{4} x + \cos^{6} x + ....... \infty) \log_{e}^{2}}$ satisfies the equation $t^{2} - 9 t + 8 = 0$ , then the value of $\frac{2 \sin x}{\sin x + \sqrt{3} \cos x} (0 < x < \frac{π}{2})$ is

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a

$2 \sqrt{3}$

b

$3 / 2$

c

$\sqrt{3}$

d

$1 / 2$

answer is D.

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Detailed Solution

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$\begin{array}{l} e^{(\cos^{2} x + \cos^{4} x + ..... \infty) \log_{e}^{} 2} \\ = 2^{\cot^{2} x} \end{array}$ $\begin{array}{l} 2^{\cot^{2} x} = 1, 8 \\ \Rightarrow \cot^{2} x = 0, 3 \end{array}$

$\begin{array}{l} t^{2} - 9 t + 8 \Rightarrow t = 1, 9 \\ \frac{2 \sin x}{\sin x + \sqrt{3} \cos x} = \frac{2}{1 + \sqrt{3} \cot x} = \frac{2}{4} = \frac{1}{2} \end{array}$

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