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$$\tan 20^{\circ} = x \Rightarrow \frac{\tan 259^{\circ} + \tan 340^{\circ}}{\tan 200^{\circ} - \tan 110^{\circ}}$$
$$\frac{1 + x^{2}}{1 - x^{2}}$$
$$\frac{1 + x}{1 - x}$$
$$\frac{1 - x^{2}}{1 + x^{2}}$$
$$\frac{1 - x}{1 + x}$$
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