To Prove cos(3x) = 4cos3(x) 3cos(x).
Use the angle sum formula for cos(3x) = cos(2x + x)
cos(3x) = cos(2x) cos(x) sin(2x) sin(x)
Substitute cos(2x) = 2cos2(x) 1 and sin(2x) = 2sin(x) cos(x):
cos(3x) = (2cos2(x) 1) cos (x) – (2sin(x) cos(x)) sin(x).
cos(3x) = 2cos3(x) cos(x) 2sin2(x) cos(x).
Use sin2(x) = 1 cos2(x):
cos(3x) = 2cos3(x) cos(x) 2(1cos2(x))cos(x).
cos(3x) = 2cos3(x) cos(x) 2cos(x) + 2cos3(x).
cos(3x) = 4cos3(x) 3cos(x)
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