cos(2θ) = 2cos2(θ) - 1
cos(2θ) = 1 - 2sin2(θ)
Reverse these to derive the half angle formulas.
cos(½θ) = sqrt(½(1+cos(θ)))
sin(½θ) = sqrt(½(1-cos(θ)))
Notice that sine squared + cosine squared is still 1, as required.
Let's try 15°, which is half of 30°, which has a cosine of sqrt(3)/2. After some algebra,
cos(15°) = sqrt(2+sqrt(3))/2 = 0.9659
sin(15°) = sqrt(2-sqrt(3))/2 = 0.2588
In this case we could have derived the sine and cosine via angle subtraction. That is, cos(45°-30°) = sqrt(1/2)×(1/2+sqrt(3)/2). Oddly enough, this different looking formula produces the exact same number.
As an exercise, use the half angle formula to show the tangent of 22.5 degrees is sqrt(2)-1.