By the angle difference formula for sines, we have:
[tex]\sin 15^\circ = \sin 45^\circ \cos 30^\circ - \sin 30^\circ \cos 45^\circ = \frac{sqrt{6}}{4} - \frac{sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.[/tex]
By the angle sum formula for tangents, we have:
[tex]\tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} = \frac{1+\frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}}[/tex].
Rationalizing the denominator gives [tex]\frac{12+6\sqrt{3}}{6} = 2+\sqrt{3}[/tex] as the final answer.
