330/2=165 and 330=360-30. The angles 30 and 360 have sines and cosines we know the values of: sin360=0, cos360=1, sin30=1/2, cos30=√3/2. Also cos330=cos(-30)=cos(30)=√3/2, because -30 and 30 are in quadrants IV and I where cosine is positive.
CosA=2cos^2(A/2)-1, so cos^2(A/2)=½(cosA+1) and cos(A/2)=√(½(cosA+1)).
Let A=330, then cos(165)=√(½(cos330+1))=√(½(cos30+1))=√(½(√3/2+1))=½√(2+√3).
But 165 is in the second quadrant where cos is negative, making the true answer -½√(2+√3).
This can also be written -(√6+√2)/4, because, if we square this we get (6+2+2√12)/16=(8+4√3)/16=(2+√3)/4. If we now take the square root of the latter we get ±½√(2+√3). The negative value corresponds to the result we got earlier. The simplest answer then is -(√6+√2)/4 because it does not involve a double square root.