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CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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sin⁡(3π/2)=−1\sin(3\pi/2)=-1 shows that the positive square root is not an unconditional half-angle formula

Statement refuted

The unconditional assertion sin⁡(x/2)=(1−cos⁡x)/2\sin(x/2)=\sqrt{(1-\cos x)/2} for every real xx is false. The conventions and prerequisite facts used below are recorded in Half-angle identities with the sign determined by the quadrant, Quarter-turn values and shifts by pi/2 and pi.

Facts & Assumptions

Given: x=3πx=3\pi.

Counterexample

1.1

Here sin⁡(x/2)=sin⁡(3π/2)=−1\sin(x/2)=\sin(3\pi/2)=-1, while (1−cos⁡3π)/2=1\sqrt{(1-\cos3\pi)/2}=1.

given
2.1

Thus the positive square root loses the required quadrant sign.

algebra∎

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