use the sum to product formula to find the exact value of

Sum to product formula for cosine function is

cos(u) - cos (v) = -2sin((u+v)/2).sin((u-v)/2)

substitute u = 165 and v = 75 in above equation (assume all angles are in degrees)

cos165 - cos75

= -2sin((165+75)/2).sin((165-75)/2)

= -2sin(240/2).sin(90/2)

= -2sin(120)sin(45)

= -2 (sqrt(3)/2)(sqrt(2)/2)

= -sqrt(6)/2

= -1.225