<s>[INST] How many tangent half-angle formulas are there? \begin{align}
\tan \frac{\alpha+\beta} 2 & = \frac{\sin\alpha+\sin\beta}{\cos\alpha + \cos\beta} \tag 1 \\[10pt]
\tan \left( \frac \pi 4 \pm \frac \alpha 2 \right) & = \sec\alpha \pm \tan\alpha \tag 2 \\[10pt]
\frac{1 + i\tan\frac\alpha2}{1-i\tan\frac\alpha2} & = e^{i\alpha} \tag 3 \\[10pt]
\tan\frac\alpha2\cdot\tan\frac\beta2 = \tan\frac\gamma2 & \text{ if and/or only if } \left|\frac{\cos\alpha+\cos\beta}{1+\cos\alpha\cos\beta}\right| = \left|\cos\gamma\right| \tag 4
\end{align}
All of these relate tangents of half angles to trigonometric functions of whole angles.
Are there other tangent half-angle formulas essentially different from these?  And how does one decide whether they're essentially different? (I'm not sure I would count the case of $(1)$ in which $\alpha=0$ as essentially different. One could make a case that $(1)$ is not essentially different from $(2)$.  We're all accustomed to the mapping $(x,y)\mapsto \dfrac{x+y}{1-xy}$ in connection with tangents, but $(4)$ relates tangents to $(x,y)\mapsto\dfrac{x+y}{1+xy}.$)
 [/INST] For $\alpha+\beta+\gamma=\pi$,
\begin{align}
\tan\tfrac\alpha2\,\tan\tfrac\beta2\tan\tfrac\gamma2
&=
\frac{\cos\alpha+\cos\beta+\cos\gamma-1}{\sin\alpha+\sin\beta+\sin\gamma}.
\end{align}
Edit
Some more
\begin{align}
\tan\tfrac\alpha2\,\tan\tfrac\beta2\tan\tfrac\gamma2
&=
\frac{2\sin\alpha\sin\beta\sin\gamma}{(\sin\alpha+\sin\beta+\sin\gamma)^2}
\\
&=
\frac{\sin2\alpha+\sin2\beta+\sin2\gamma}{2(\sin\alpha+\sin\beta+\sin\gamma)^2}
\end{align}
\begin{align}
\cot\tfrac\alpha2+\cot\tfrac\beta2+\cot\tfrac\gamma2-(\tan\tfrac\alpha2+\tan\tfrac\beta2+\tan\tfrac\gamma2)
&=
2(\cot\alpha+\cot\beta+\cot\gamma)
\end{align}
</s>