$\quad \begin{cases}\tan(a+b)=5\\\tan(a-b)=3\end{cases}$
$\Leftrightarrow \begin{cases}a + b =\arctan5\\a-b=\arctan3\end{cases}$
$\Leftrightarrow \begin{cases}2a = \arctan5 + \arctan3\\b = a - \arctan3\end{cases}$
$\Leftrightarrow \begin{cases}a =\dfrac{\arctan5 +\arctan3}{2}\\b =\dfrac{\arctan5 - \arctan3}{2}\end{cases}$