Đáp án:
$\begin{array}{l}
\sin x = \dfrac{1}{3}\\
Do:{\cot ^2}x + 1 = \dfrac{1}{{{{\sin }^2}x}}\\
\Rightarrow {\cot ^2}x + 1 = 9\\
\Rightarrow {\cot ^2}x = 8\\
\Rightarrow \cot x = 2\sqrt 2 \\
\Rightarrow \tan x = \dfrac{1}{{\cot x}} = \dfrac{1}{{2\sqrt 2 }}\\
A = \dfrac{{\tan x + 3\cot x}}{{\tan x + \cot x}}\\
= \dfrac{{\dfrac{1}{{2\sqrt 2 }} + 3.2\sqrt 2 }}{{\dfrac{1}{{2\sqrt 2 }} + 2\sqrt 2 }}\\
= \dfrac{{\dfrac{{1 + 3.2\sqrt 2 .2\sqrt 2 }}{{2\sqrt 2 }}}}{{\dfrac{{1 + 2\sqrt 2 .2\sqrt 2 }}{{2\sqrt 2 }}}}\\
= \dfrac{{1 + 3.8}}{{1 + 8}} = \dfrac{{25}}{9}
\end{array}$
