SOLUTION: prove sin 75 degrees – sin 15 degrees = cos 105 degrees + cos 15 degrees

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Question 202605: prove sin 75 degrees – sin 15 degrees = cos 105 degrees + cos 15 degrees
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Rewrite sin(75) and cos(105) as sin(90-15) and cos(90+15):
sin(90-15) - sin(15) = cos(90+15) + cos(15)

Using the sin(a-b) = sin(a)cos(b) - cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b) formulas we get:
sin(90)cos(15) - cos(90)sin(15) - sin(15) = cos(90)cos(15) - sin(90)sin(15) + cos(15)

sin(90) = 1
cos(90) = 0
Substituting these into our equation we get:
1*cos(15) - 0*sin(15) - sin(15) = 0*cos(15) - 1*sin(15) + cos(15)
Simplifying we get:
cos(15) - 0 - sin(15) = 0 - sin(15) + cos(15)
or
cos(15) - sin(15) = cos(15 - sin(15)