Vectors & Tetrahedrons: Show that $P^2 = B^2 + R^2 + G^2 + 2BR \cos ∠(b, r) + 2RG \cos ∠(r, g) + 2GB \cos ∠(g, b).$

Question

Show that $$P^2 = B^2 + R^2 + G^2 - 2BR \cos ∠(B, R) - 2RG \cos ∠(R, G) - 2GB \cos ∠(G, B).$$ Here, ∠(B, R) denotes the angle between the two faces coloured blue and red, and similarly for the other two angles in this formula. This formula is a three-dimensional counterpart of a very famous trigonometric identity for triangles. Which identity is that?

I don't know how to go about this question, this is a part of a long answer question, can anyone please help me through this please!

Answer 1

This is Al-Kashi theorem (or the law of cosines) in the euclidean space. You can check p. 133 of this book :

Casey, John (1889). A Treatise on Spherical Trigonometry: And Its Application to Geodesy and Astronomy with Numerous Examples. London: Longmans, Green, & Company.