2. The smallest positive integer $x$ that satisfies the equation $\tan 19 x^{\circ}=\frac{\cos 99^{\circ}+\sin 99^{\circ}}{\cos 99^{\circ}-\sin 99^{\circ}}$ is $x=$ $\qquad$ .
2.36.

The right side of the original equation $=\frac{1+\tan 99^{\circ}}{1-\tan 99^{\circ}}=\frac{\tan 45^{\circ}+\tan 99^{\circ}}{1-\tan 45^{\circ} \cdot \tan 99^{\circ}}$ $=\tan 144^{\circ}$.

Then $19 x=144+180 k(k \in \mathbf{N})$.
So $x=\frac{180 k+144}{19}=9 k+7+\frac{9 k+11}{19}$.
We know $19 \mid 9 k+11$. Thus, $k_{\operatorname{man}}=3$, hence,
$$
\begin{array}{l}
x=9 \times 3+7+ \\
\frac{9 \times 3+11}{19}=36 .
\end{array}
$$