The identity for the half angle of the cosine is:
cos(x/2) = +-sqrt((1+cos(x))/2cos(x2)=±√1+cos(x)2
We are about to substitute x/2 = (11pi)/12x2=11π12 but, before we do that, we must observe that (11pi)/1211π12 is in the second quadrant. The cosine function is negative in the second quadrant, therefore, we must use the negative value:
cos(x/2) = -sqrt((1+cos(x))/2cos(x2)=−√1+cos(x)2
Substitute x/2 = (11pi)/12x2=11π12 and x = (11pi)/6x=11π6 into the identity:
cos((11pi)/12) = -sqrt((1+cos((11pi)/6))/2cos(11π12)=−
⎷1+cos(11π6)2
Substitute cos((11pi)/6)= sqrt3/2cos(11π6)=√32:
cos((11pi)/12) = -sqrt(((1+sqrt3/2))/2)cos(11π12)=−
⎷(1+√32)2
cos((11pi)/12) = -sqrt((((2+sqrt3)/2))/2)cos(11π12)=−
⎷(2+√32)2
cos((11pi)/12) = -sqrt((2+sqrt3)/4)cos(11π12)=−√2+√34
cos((11pi)/12) = -sqrt(2+sqrt3)/2cos(11π12)=−√2+√32
