<s>[INST] Calculus sine proof Suppose that $a, b, c$ are non-zero acute angles such that
$$\frac{\sin(a − b)}{\sin(a + b)} + \frac{\sin(b − c)}{\sin(b + c)} + \frac{\sin(c − a)}{\sin(c + a)}= 0$$
Prove that at least two of $a, b, c$ are equal.
I have no idea how to begin. 
 [/INST] $$\frac{\sin(a-b)}{\sin(a+b)}=\frac{\alpha/\beta-\beta/\alpha}
{\alpha\beta-1/(\alpha\beta)}=\frac{\alpha^2-\beta^2}{\alpha^2\beta^2-1}$$
where $\alpha=\exp(ia)$ and $\beta=\exp(ib)$. Set $\gamma=\exp(ic)$.
Then
$$(\alpha^2-\beta^2)(\alpha^2\gamma^2-1)(\beta^2\gamma^2-1)
+(\beta^2-\gamma^2)(\alpha^2\beta^2-1)(\alpha^2\gamma^2-1)
+(\gamma^2-\alpha^2)(\alpha^2\beta^2-1)(\beta^2\gamma^2-1)
=0.$$
This simplifies to
$$(\alpha^4-\beta^4)\gamma^2+(\beta^4-\gamma^4)\alpha^2
+(\gamma^4-\alpha^4)\beta^2=0.$$
But that factors as
$$(\alpha^2-\beta^2)(\beta^2-\gamma^2)(\gamma^2-\alpha^2)=0.$$
If say $\alpha^2=\beta^2$ then $a=b$, as $a$ and $b$ are acute angles.
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