Step-by-step Solution:

Eliminating \( y \) from the given relations: \[ \cos x = \tan y = \frac{1}{\cot y} = \frac{1}{\tan z} = \cot z = \tan x \] \[ \Rightarrow \cos x = \tan x \] Or: \[ \sin x = \cos^2 x = 1 - \sin^2 x \] \[ \Rightarrow \sin x = \frac{-1 + \sqrt{5}}{2} \]