Mathematics - Sequence & Series Question with Solution | TestHub

$e^{\left({\cos }^{2}x+{\cos }^{4}x+\dots \dots \infty \right)\ln 2}=2^{{\cos }^{2}x+{\cos }^{4}x+\dots \dots \infty }$

$==2^{\frac{{\cos }^{2}x}{1-{\cos }^{2}x}}=2^{{\cot }^{2}x}$

$t^2-9t+8=0\Rightarrow t=1,8$

$\Rightarrow 2^{{\cot }^{2}x}=1, 8\Rightarrow {\cot }^{2}x=0,3$

$0<x<\frac{\pi }{2}\Rightarrow \cot x=\sqrt{3}$

$\Rightarrow \frac{2 \sin x}{\sin x+\sqrt{3} \cos x}=\frac{2}{1+\sqrt{3} \cot x}=\frac{2}{4}=\frac{1}{2}$