[tex]\sin^2 \frac{\alpha }{2} = \frac{1-\cos \alpha }{2} \\\sin^2 15^{\circ} = \frac{1-\cos 30^{\circ} }{2} =\sin^2 15^{\circ} = \frac{1-\frac{\sqrt 3}{2} }{2}=\frac{2-\sqrt 3}{4} \\\sin 15^{\circ} = \frac{\sqrt{2-\sqrt 3} }{2}\\\sin 30^{\circ} \sin 15^{\circ} = \frac{\sqrt{2-\sqrt 3} }{2} \cdot \frac{1}{2} = \frac{\sqrt{2-\sqrt 3} }{4}[/tex]