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

Solution:

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 

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