The identity for the half angle of the cosine is:
#cos(x/2) = +-sqrt((1+cos(x))/2#
We are about to substitute #x/2 = (11pi)/12# but, before we do that, we must observe that #(11pi)/12# 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))/2#
Substitute #x/2 = (11pi)/12# and #x = (11pi)/6# into the identity:
#cos((11pi)/12) = -sqrt((1+cos((11pi)/6))/2#
Substitute #cos((11pi)/6)= sqrt3/2#:
#cos((11pi)/12) = -sqrt(((1+sqrt3/2))/2)#
#cos((11pi)/12) = -sqrt((((2+sqrt3)/2))/2)#
#cos((11pi)/12) = -sqrt((2+sqrt3)/4)#
#cos((11pi)/12) = -sqrt(2+sqrt3)/2#
