Đáp án:
$\begin{array}{l}
0 < a < {90^0}\\
\Rightarrow \sin a > 0\\
\Rightarrow \left\{ \begin{array}{l}
\tan a > 0\\
\cot a > 0
\end{array} \right.\\
Do:{\sin ^2}a + {\cos ^2}a = 1\\
\Rightarrow {\sin ^2}a = 1 - {\cos ^2}a\\
= 1 - {\left( {\dfrac{3}{4}} \right)^2} = \dfrac{7}{{16}}\\
\Rightarrow \sin a = \dfrac{{\sqrt 7 }}{4}\\
\Rightarrow \left\{ \begin{array}{l}
\tan a = \dfrac{{\sin a}}{{\cos a}} = \dfrac{{\dfrac{{\sqrt 7 }}{4}}}{{\dfrac{3}{4}}} = \dfrac{{\sqrt 7 }}{3}\\
\cot a = \dfrac{1}{{\tan a}} = \dfrac{3}{{\sqrt 7 }}
\end{array} \right.\\
A = \dfrac{{\tan a + 3\cot a}}{{\tan a + \cot a}}\\
= \dfrac{{\dfrac{{\sqrt 7 }}{3} + 3.\dfrac{3}{{\sqrt 7 }}}}{{\dfrac{{\sqrt 7 }}{3} + \dfrac{3}{{\sqrt 7 }}}} = \dfrac{{\dfrac{{7 + 27}}{{3\sqrt 7 }}}}{{\dfrac{{7 + 9}}{{3\sqrt 7 }}}} = \dfrac{{34}}{{16}} = \dfrac{{17}}{8}\\
Vay\,A = \dfrac{{17}}{8}
\end{array}$
