Find the Sum

Geometry Level 3

\[ \large \dfrac x{\cos \theta} = \dfrac y{\cos \left( \theta + \frac{2\pi}3 \right) }= \dfrac z{\cos \left( \theta - \frac{2\pi}3 \right) } \]

If the equation above is an identity except when \( \cos \theta, \cos \left( \theta + \frac{2\pi}3 \right), \cos \left( \theta - \frac{2\pi}3 \right) \ne 0 \), find \(x+y+z\).

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