$解：(1) 因为 \cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta.$
$所以 \cos 75^{\circ}=\cos (30^{\circ}+45^{\circ})= \cos 30^{\circ} \cos 45^{\circ}-\sin 30^{\circ} \sin 45^{\circ}$
$=\frac {\sqrt{3}}{2} \times \frac {\sqrt{2}}{2}-\frac {1}{2} \times \frac {\sqrt{2}}{2}=\frac {\sqrt{6}-\sqrt{2}}{4}.$
$(2)因为 \tan \alpha=\frac {\sin \alpha}{\cos \alpha}，$
$所以 \tan 75^{\circ}=\frac {\sin 75^{\circ}}{\cos 75^{\circ}}=\frac {\frac {\sqrt{2}+\sqrt{6}}{4}}{\frac {\sqrt{6}-\sqrt{2}}{4}}=2+\sqrt{3} .$
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