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cos 165 degrees - cos 75 degrees

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

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Rajesh P. | Effective Tutor for Test Prep, Math, Science, Comp Science, BusinessEffective Tutor for Test Prep, Math, Sci...
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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