By the tangent addition formula, \begin{align*} \frac{\cos 96^\circ + \sin 96^\circ}{\cos 96^\circ - \sin 96^\circ} &= \frac{1 + \tan 96^\circ}{1 - \tan 96^\circ} \\ &= \frac{\tan 45^\circ + \tan 96^\circ}{1 - \tan 45^\circ \tan 96^\circ} \\ &= \tan (45^\circ + 96^\circ) \\ &= \tan 141^\circ.\end{align*}Thus, we seek the smallest positive integer solution to \[\tan 19x^\circ = \tan 141^\circ.\]This means $ 19x - 141 = 180n $ for some integer $ n, $ or $ 19x - 180n = 141 $. We can use the Extended Euclidean Algorithm to find the smallest positive integer solution. Running the Euclidean Algorithm on 180 and 19, we get \begin{align*} 180 &= 9 \cdot 19 + 9, \\ 19 &= 2 \cdot 9 + 1, \\ 9 &= 9 \cdot 1.\end{align*}Then \begin{align*} 1 &= 19 - 2 \cdot 9 \\ &= 19 - 2 \cdot (180 - 9 \cdot 19) \\ &= 19 \cdot 19 - 2 \cdot 180.\end{align*}Multiplying both sides by 141, we get \[2679 \cdot 19 - 282 \cdot 180 = 141.\]Note that if $ (x,n) $ is a solution to $ 19x - 180n = 141, $ then so is $ (x - 180,n + 19) $. Thus, we reduce 2679 modulo 180, to get $ x = \boxed{159} $. Alternatively, we want to solve \[19x \equiv 141 \pmod{180}.\]Multiplying both sides by 19, we get \[361x \equiv 2679 \pmod{180},\]which reduces to $ x \equiv \boxed{159} \pmod{180} $.