A new trig identity

This evening I ran across a trig identity I hadn’t seen before. I doubt it’s new to the world, but it’s new to me.

Let A, B, and C be the angles of an arbitrary triangle. Then

sin² A + sin² B + sin² C = 2 + 2 cos A cos B cos C.

This looks a little like the Pythagorean theorem, but the Pythagorean theorem involves the sides of a triangle, not the angles. (I expect there’s an interesting generalization of the identity above to spherical geometry where sides are angles.)

This identity also looks a little like the law of cosines, but the law of cosines mixes sides and angles, and this identity only involves angles.

Source: Lemma 4.4.3 in A Panoply of Polygons by Claudi Alsina and Roger B. Nelsen.

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