Example 8 (10th "Hope Cup" Invitational Competition Question) 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 $x=$ $\qquad$ .
Solve: Fill in 36. Reason: Since $\tan 19 x^{\circ}=\frac{-\sin 9^{\circ}+\sin 99^{\circ}}{-\sin 9^{\circ}-\sin 99^{\circ}}=\frac{\sin 9^{\circ}-\sin 99^{\circ}}{\sin 9^{\circ}+\sin 99^{\circ}}$
$$
=-\frac{2 \cos 54^{\circ} \cdot \sin 45^{\circ}}{2 \sin 54^{\circ} \cdot \cos 45^{\circ}}=-\cot 54^{\circ}=\tan 144^{\circ} .
$$

Therefore, $19 x=144+180 \cdot k, k \in \mathbf{N}$, hence, $x=\frac{180 k+144}{19}=9 k+7+\frac{9 k+11}{19}$,
Since $19 \mid(9 k+11)$, then $k_{\text {min}}=3$.
Thus, $x=9 \cdot 3+7+\frac{9 \cdot 3+11}{19}=36$.