Cosine-Sine Angle Rule

Around 1998, I conjectured the following trigonometric result on the basis of the side-angle duality that often appears in Euclidean geometry. It was specifically inspired by the (not so well-known, except in problem-solving circles) trigonometric version of Ceva's theorem. The validity of this conjecture was quickly confirmed by construction and dragging with the dynamic geometry software Sketchpad, as well as by a one line proof. Though this curious-looking result is in all likelihood not new or original, it doesn't seem to be well known or appear in standard mathematical textbooks.

Result: In any triangle ABC, (sin A)2 = (sin B)2 + (sin C)2 - 2 sin B sin C cos A.


Cosine-Sine Angle Rule

Explanation (proof):
Can you explain why (prove) this result true? Click on the provided HINT in the sketch if you get stuck.

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Created online by Michael de Villiers, around 2005, updated 25 May 2017.