The cosine half-angle formula is
#cos(x/2) = ±sqrt((1 + cos x) / 2)#
The sign is positive if #x/2# is in the first or fourth quadrant and negative if #x/2# is in the second or third quadrant.
#-(5π)/12# is in the fourth quadrant, so the sign is positive.
Also, #cos(-x) = cosx#, and
#(5π)/12 = 1/2×(5π)/6#
∴ #cos(-(5π)/12) =cos((5π)/12) = cos(((5π)/6)/2) = sqrt((1+cos ((5π)/6))/2)#
#cos(-(5π)/12) =sqrt((1+cos (π - π/6))/2) == sqrt((1 – cos(π/6))/2)#
#cos(-(5π)/12) = sqrt((1 – (sqrt3)/2)/2) = sqrt((2 - sqrt3)/4)#
#cos(-(5π)/12) = sqrt(2 - sqrt3)/2#
