If $\cos x= an y, \cot y= an z$ and $\cot z= an x$, then $\sin x$ equals to

Question

If $\cos x=	an y, \cot y=	an z$ and $\cot z=	an x$, then $\sin x$ equals to

Aicharya · Accepted Answer

Given, $\cos x=	an y, \cot y=	an z$ and $\cot z=	an x$ $	herefore \quad \cos x=	an y$ $\Rightarrow \quad \cos x=\frac{1}{	an z}$ $\Rightarrow \quad \cos x=\cot z$ $\Rightarrow \quad \cos x=	an x$ $\Rightarrow \quad \cos x=\frac{\sin x}{\cos x}$ $\Rightarrow \cos ^2 x=\sin x$ $\Rightarrow 1-\sin ^2 x=\sin x$ $\Rightarrow \sin ^2 x+\sin x-1=0$ $\begin{aligned} 	herefore \quad \sin x & =\frac{-1 \pm \sqrt{1-4 	imes(-1)}}{2 	imes 1} \ & =\frac{-1 \pm \sqrt{5}}{2} \ 	herefore \sin x=\frac{\sqrt{5}-1}{2} & \left(\because \frac{-1-\sqrt{5}}{2} < -1ight)\end{aligned}$

If $\cos x=\tan y, \cot y=\tan z$ and $\cot z=\tan x$, then $\sin x$ equals to

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