It's a trigonometric dilemma

Geometry Level 4

\[\large \begin{cases} \cos x =\tan y \\ \cos y = \tan z \\ \cos z =\tan x \end{cases}\]

If \( x, y\) and \(z\) are real numbers that satisfy the above system of equations with \( 0 \leq \sin x \leq 1 \), then what is the value of \( \sin x \)?

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