S = cos ((5pi)/12) + cos ((2pi)/3 - pi/4)
cos ((2pi)/3 - pi/4) = cos ((8pi)/12 - ((3pi)/12) = cos ((5pi)/12)
Trig unit circle and property of complementary arcs give -->
cos ((5pi)/12) = cos (pi/2 - pi/12) = sin (pi/12)
Therefor,
S = 2sin (pi/12)
Evaluate sin (pi/12) by the trig identity: cos 2a = 1 - 2sin^2 a
cos (pi/6) = sqrt3/2 = 1 - 2sin^2 (pi/12)
2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2
sin^2 (pi/12) = (2 - sqrt3)/4
sin (pi/12) = sqrt(2 - sqrt3)/2 --> sin (pi)/12 is positive
S = 2sin (pi/12) = sqrt((2 - sqrt3)
