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

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$\frac{\sqrt{5} + 1}{2}$

$\frac{\sqrt{5} - 1}{2}$

$\frac{\sqrt{5} + 1}{4}$

$\frac{\sqrt{5} - 1}{4}$

answer is D.

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Detailed Solution

$\begin{array}{r} \begin{matrix} \cos x = \tan y \\ \tan z = \cot y \\ \cot z = \tan x \end{matrix} \end{array}\}$ multiplying we get $\cos x = \tan x$

$\Rightarrow \cos^{2} x = \sin x$

$\Rightarrow \sin^{2} x + \sin x - 1 = 0 \Rightarrow \sin x = \frac{- 1 \pm \sqrt{5}}{2}$

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