cos(3x) = cos(2x)cos(x) - sin(2x)sin(x)
= [cos(x)cos(x) - sin(x)sin(x)]cos(x) - [sin(x)cos(x) + sin(x)cos(x)]sin(x)
= [cos^2(x) - sin^2(x)]cos(x) - [2sin(x)cos(x)]sin(x)
= cos^3(x) - sin^2(x)cos(x) - 2sin^2(x)cos(x)
= cos^3(x) - 3sin^2(x)cos(x).
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This is accomplished by using algebra, and two rules:
(1) cos(a+b)= cos(a)cos(b) - sin(a)sin(b)
(2) sin(a+b) = sin(a)cos(b) - sin(b)cos(a)
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(You can ask questions in the comments)
