Half-Angle Values

Put x/2 in place of x in the power-reducing formulas and they find the values at half an angle you know:

sin(x/2)² = (1 - cos(x))/2

cos(x/2)² = (1 + cos(x))/2

15° is half of 30°. So sin(15°)² = (1 - cos(30°))/2 = (1 - √(3)/2)/2. Take the half out of the top: (1/2)(2 - √(3))/2, which is (2 - √(3))/4.

The formula gives the square. To get sin 15° itself take the square root and choose the sign from the quadrant: 15° is in quadrant I, so the positive root, sin(15°) = √(2 - √(3))/2.