Trig unit circle -->
#cos ((11pi)/12) = cos (- pi/12 + pi) = - cos (pi/12)# (1)
Evaluate #cos (pi/12)# by the trig identity:
#2cos^2a = 1 + cos 2a#
#2cos ^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 (pi/12) = (2 + sqrt3)/4#
#cos (pi/12) = +- sqrt(2 + sqrt3)/2#
Since #cos (pi/12)# is positive, then only the positive value is accepted.
Finally, from (1):
#cos ((11pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2#
Check by calculator.
#cos ((11pi)/12) = cos 165 = - 0.97#
# - sqrt(2 + sqrt3)/2 = sqrt(3.73)/2 = 1.93/2 = 0.97.# OK