Answer: A

Solution:

Since $\cos x = 0,$ we have $x = \tfrac{\pi}{2} + k\pi.$ Since $\cos(x + z) = \tfrac{1}{2},$ we have $x + z = 2n\pi \pm \tfrac{\pi}{3}.$

On the unit circle, the smallest positive angular separation between an odd multiple of $\tfrac{\pi}{2}$ and an angle congruent to $\pm\tfrac{\pi}{3}$ is $\tfrac{\pi}{6}.$ It is attained by taking $x = -\tfrac{\pi}{2}$ and $x + z = -\tfrac{\pi}{3},$ which gives $z = -\tfrac{\pi}{3} + \tfrac{\pi}{2} = \tfrac{\pi}{6}.$

Thus, the correct answer is A.