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$z = 90^\circ$: $\sin(180^\circ) = 0$, $\cos(90^\circ) = 0$ → 0 = 0 → OK.
$z = 270^\circ$: $\sin(540^\circ) = \sin(180^\circ) = 0$, $\cos(270^\circ) = 0$ → 0 = 0 → OK.
So all four satisfy? But earlier algebra says $\cos z (2\sin z - 1) = 0$, so $\cos z = 0$ or $\sin z = 1/2$.

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