Answer: 159

Small Hint:

Recognize the right-hand side using the tangent addition formula

Big Hint:

Solve the resulting congruence modulo $180$

Solution:

The tangent addition formula gives $$\begin{gathered}\frac{\cos96^\circ+\sin96^\circ}{\cos96^\circ-\sin96^\circ}\\=\tan(45^\circ+96^\circ)\\=\tan141^\circ.\end{gathered}$$ Hence $19x\equiv141\pmod {180}.$ Since $19^2=361\equiv1\pmod {180},$ multiplying by $19$ gives $$x\equiv19\cdot141\equiv159\pmod {180}.$$ The smallest positive solution is $159.$