If $\frac{\mathrm{x}}{\cos \mathrm{\theta }}=\frac{\mathrm{y}}{\cos \left(\mathrm{\theta }-\frac{2\mathrm{\pi }}{3}\right)}=\frac{\mathrm{z}}{\cos \left(\mathrm{\theta }+\frac{2\mathrm{\pi }}{3}\right)}$, then x + y + z is equal to

We have

$\frac{\mathrm{x}}{\cos \mathrm{\theta }}=\frac{\mathrm{y}}{\cos \left(\mathrm{\theta }-\frac{2\mathrm{\pi }}{3}\right)}=\frac{\mathrm{z}}{\cos \left(\mathrm{\theta }+\frac{2\mathrm{\pi }}{3}\right)}$

Therefore, each ratio is equal to

$\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{\cos \mathrm{\theta }+\cos \left(\mathrm{\theta }-\frac{2\mathrm{\pi }}{3}\right)+\cos \left(\mathrm{\theta }+\frac{2\mathrm{\pi }}{3}\right)}=\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{0}$

or     x + y + z = 0