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.

it’s a true trig formula but not new :-( , see wikipedia:

https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Further_%22conditional%22_identities_for_the_case_%CE%B1_+_%CE%B2_+_%CE%B3_=_180%C2%B0