Duarti wrote:
Solve the system of equations
$x+y+z=\pi$
$\tan x\tan z=2$
$\tan y\tan z=18$
Let $u=\tan x$, $v=\tan y$ and $w=\tan z$ so that system is :
$w=\frac{u+v}{uv-1}$ and $uw=2$ and $vw=18$
With easy solutions $\left(\frac 12,\frac 92,4\right)$ and $\left(-\frac 12,-\frac 92,-4\right)$
And so :
$\boxed{\text{S1 : }(x,y,z)=(\arctan \frac 12+m\pi,\arctan \frac 92+n\pi,\arctan 4-(m+n)\pi)}$, whatever are $m,n\in\mathbb Z$
$\boxed{\text{S2 : }(x,y,z)=(-\arctan \frac 12+m\pi,-\arctan \frac 92+n\pi,-\arctan 4-(m+n-2)\pi)}$, whatever are $m,n\in\mathbb Z$