If $\cos x=\tan y;\cot y=\tan z;\cot z=\tan x$ then $\sin x=$

$\cos x=\tan y\to \left(1\right);\tan z=coty\to \left(2\right);\cot z=\tan x\to \left(3\right)$

(1)x(2)x(3)
$Tanx=\cos x$
$\frac{\sin x}{\cos x}=\cos x$
$\sin x={\cos }^{2}x$
$\Rightarrow \sin x=1-{\sin }^{2}x$
${\sin }^{2}x+\sin x-1=0$
$a=1,b=1,c=-1$
$\sin x=\frac{-1\pm \sqrt{1-4\left(1\right)}\left(-1\right)}{2}$
$\sin x=\frac{-1\pm \sqrt{5}}{2}$
$\sin x=\frac{\sqrt{5}-1}{2}$
$\sin x=2\left(\frac{\sqrt{5}-1}{4}\right)$
$\sin x=2\sin {18}^{\circ }$