$\text{If }\cos \mathrm{x}=\tan \mathrm{y},\cot \mathrm{y}=\tan \mathrm{z}\text{ and }\cot \mathrm{z}=\tan \mathrm{x}\text{; then }\sin \mathrm{x}=$

$$\begin{gathered} \cos \mathrm{x}=\tan \mathrm{y}=\frac{1}{\cot \mathrm{y}}=\frac{1}{\tan \mathrm{z}}=\cot \mathrm{z}=\tan \mathrm{x} \\ \begin{array}{l}\Rightarrow \cos \mathrm{x}-\tan \mathrm{x}=0\Rightarrow {\cos }^2\mathrm{x}-\sin \mathrm{x}=0\Rightarrow 1-{\sin }^2\mathrm{x}-\sin \mathrm{x}=0\\ \Rightarrow {\sin }^2\mathrm{x}+\sin \mathrm{x}-1=0\Rightarrow \sin \mathrm{x}=\frac{-1\pm \sqrt{1+4}}{2}=\frac{-1\pm \sqrt{5}}{2}\Rightarrow \sin \mathrm{x}=\frac{\sqrt{5}-1}{2}\end{array} \end{gathered}$$