Let A and B denote the statements: A: ++=0; B: ++=0.

Question

Let $A$ and $B$ denote the statements:
$A$: $\cos\alpha+\cos\beta+\cos\gamma=0$; $B$: $\sin\alpha+\sin\beta+\sin\gamma=0$.

If $\cos(\beta-\gamma)+\cos(\gamma-\alpha)+\cos(\alpha-\beta) = -\dfrac32$, then:

Accepted Answer

$$2[\cos(\beta-\gamma)+\cos(\gamma-\alpha)+\cos(\alpha-\beta)]+3=0$$

Adding $\sin^2\alpha+\cos^2\alpha+\sin^2\beta+\cos^2\beta+\sin^2\gamma+\cos^2\gamma=3$ and expanding each $\cos(X-Y)=\cos X\cos Y+\sin X\sin Y$:

$$(\sin\alpha+\sin\beta+\sin\gamma)^2 + (\cos\alpha+\cos\beta+\cos\gamma)^2 = 0$$

A sum of two squares equal to zero forces both to vanish, so $A$ and $B$ are both true.

Solutions · 1