Consider the following trigonometric identity, valid for any set of angles $u,v,t$:
$$\cos t⋅\cos u⋅\cos v =\frac14\left[\cos(t + u + v)+\cos(t + u - v)+\cos(u+v-t)+\cos(v + t - u)\right]$$
This identity and its derivation have previously appeared as a question on this site (though the version quoted in the question is in fact missing a term).
But the solutions to the linked question, while valid, were all algebraic in nature. Is there a geometric proof/demonstration of the above identity? I would also be satisfied with a proof-without-words of a special case e.g. $u=u+v$, $u+v+t=\pi$, $u+v+t=2\pi$, etc.