Explanation

(c)
We have,$\ x\cos{\theta = y\cos\left( \theta + \frac{2\pi}{3} \right)}$
$\mathcal{= z}\cos{\left( \theta + \frac{4\pi}{3} \right) = k}$ (say)
$\Rightarrow \cos{\theta = \frac{k}{x},\cos{\left( \theta + \frac{2\pi}{3} \right) = \frac{k}{y}}}$
and$\ \cos{\left( \theta + \frac{4\pi}{3} \right) = \frac{k}{\mathcal{z}}}$
$\therefore\ \frac{k}{x} + \frac{k}{y} + \frac{k}{\mathcal{z}} = \cos{\theta + \cos{\left( \theta + \frac{2\pi}{3} \right) + \cos\left( \theta + \frac{4\pi}{3} \right)}}$
$= \cos{\theta - \cos{\left( \frac{\pi}{3} - \theta \right) - \cos\left( \frac{\pi}{3} + \theta \right)}}$
$= \cos{\theta - 2\cos{\frac{\pi}{3}\cos{\theta = 0}}}$
$\Rightarrow \frac{1}{x} + \frac{1}{y} + \frac{1}{\mathcal{z}} = 0$