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Theta/2 is between (pi)/4 and (pi)/2 (45 and 90 degrees). This puts all trig functions in the first quadrant where they are all positive. Cosine is negative in quadrants 2 and 3 only. (Quadrant rule for quadrants 1 to 4 is All Silver Tea Cups, or All, Sine, Tangent, Cosine positive rule.) So cos(theta/2)<0 is false when (pi)/2<theta<(pi).

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