Hướng dẫn giải:

$\begin{align}& 1+\tan x=2\sqrt{2}\sin x \\& \Leftrightarrow \cos x+\sin x=\sqrt{2}\sin 2x \\& \Leftrightarrow \sin (x+\frac{\pi }{4})=\sin 2x \\& \Leftrightarrow \left[ \begin{matrix}x+\frac{\pi }{4}+k2\pi =2x  \\\frac{3\pi }{4}-x+k2\pi =2x  \\\end{matrix} \right.,k\in \mathbb{Z} \\& \Leftrightarrow \left[ \begin{matrix}x=\frac{\pi }{4}+k2\pi   \\x=\frac{\pi }{4}+k\frac{2\pi }{3}  \\\end{matrix},k\in \mathbb{Z} \right.\Rightarrow x=\frac{\pi }{4}+k\frac{2\pi }{3},k\in \mathbb{Z} \\& \Rightarrow \frac{a}{b}=\frac{1}{4} \\\end{align}$