<s>[INST] Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$ If $\alpha + \beta + \gamma = \pi $ and $\tan(\frac{-\alpha + \beta + \gamma}4)\tan(\frac{\alpha - \beta + \gamma}4)\tan(\frac{\alpha + \beta - \gamma}4) = 1$
Then prove that:

$1 + \cos\alpha + \cos\beta + \cos\gamma = 0$.

I have no idea how to go about this. 
Please help.
 [/INST] Let's introduce the shorthand notations
$$
\begin{align}
a &= \frac{-\alpha+\beta+\gamma}{4} = \frac{\pi}{4} - \frac{\alpha}{2},\\
b &= \frac{\alpha-\beta+\gamma}{4} = \frac{\pi}{4} - \frac{\beta}{2},\\
c &= \frac{\alpha+\beta-\gamma}{4} = \frac{\pi}{4} - \frac{\gamma}{2},
\end{align}
$$
then
$$
a+b = \frac{\gamma}{2},\quad a+c = \frac{\beta}{2},\quad b+c = \frac{\alpha}{2},
$$
$$
a+b+c = \frac{\pi}{4},
$$
$$
\sin a\sin b \sin c = \cos a\cos b\cos c.\tag{1}
$$
Now
$$
\cos(a+b+c) = \cos(a+b)\cos(c) - \sin(a+b)\sin c= \frac{\sqrt{2}}{2},
$$
so that
$$
\cos(a+b)\cos(c) - \sin a\sin c\cos b - \sin b\sin c\cos a = \frac{\sqrt{2}}{2}
$$
add $(2\cos a\cos b\cos c)$ to both sides, and we get
$$
\cos(a+b)\cos(c) + \cos(a+c)\cos b+ \cos(b+c)\cos(a)= \frac{\sqrt{2}}{2} + 2\cos a\cos b\cos c,
$$
which reduces to
$$
\begin{multline}
\cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\alpha}{2}\right) +
\cos\left(\frac{\beta}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\beta}{2}\right) +\\
\cos\left(\frac{\gamma}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\gamma}{2}\right) = \frac{\sqrt{2}}{2} + 2\cos a\cos b\cos c. \tag{2}
\end{multline}
$$
On the other hand,
$$
\sin(a+b+c) = \sin(a+b)\cos c + \cos(a+b)\sin c = \frac{\sqrt{2}}{2},
$$
so that, 
$$
\sin(a+b)\cos c + \cos a\cos b\sin c = \frac{\sqrt{2}}{2} + \sin a\sin b\sin c.\tag{3}
$$
Similarly, if we take $\sin(a+b+c) = \sin((a+c)+b)$, we get
$$
\sin(a+c)\cos b + \cos a\cos c\sin b = \frac{\sqrt{2}}{2} + \sin a\sin b\sin c.\tag{4}
$$
If we add eqs. $(3)$ and $(4)$, we get
$$
\sin(a+b)\cos c + \sin(a+c)\cos b + \sin(b+c)\cos a = \sqrt{2} + 2\sin a\sin b\sin c,
$$
which reduces to (using also $(1)$)
$$
\begin{multline}
\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\alpha}{2}\right) +
\sin\left(\frac{\beta}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\beta}{2}\right) +\\
\sin\left(\frac{\gamma}{2}\right)\cos\left(\frac{\pi}{4} - \frac{\gamma}{2}\right) = \sqrt{2} + 2\cos a\cos b\cos c. \tag{5}
\end{multline}
$$
Finally, we subtract eq $(5)$ from eq $(2)$:
$$
\begin{multline}
\left(\cos\left(\frac{\alpha}{2}\right)-\sin\left(\frac{\alpha}{2}\right)\right)
\cos\left(\frac{\pi}{4} - \frac{\alpha}{2}\right) + \ldots = - \frac{\sqrt{2}}{2},
\end{multline}
$$
which is
$$
\begin{multline}
\frac{\sqrt{2}}{2}\left(\cos\left(\frac{\alpha}{2}\right)-\sin\left(\frac{\alpha}{2}\right)\right)\left(\cos\left(\frac{\alpha}{2}\right)+\sin\left(\frac{\alpha}{2}\right)\right) + \ldots + \frac{\sqrt{2}}{2} =0,
\end{multline}
$$
so that finally
$$
\cos\alpha +\cos\beta+\cos\gamma + 1 = 0.
$$
</s>