Consider the well known continued fraction expansion $$ z \tan z = \frac{z^2}{1 - \cfrac{z^2}{3- \cfrac{z^2}{5 - \ldots}}} $$ of the tangent function going back to Euler and Lambert (Lambert used it for showing that $\tan z$ is irrational for rational nonzero values of $z$, which implies the irrationality of $\pi$; Legendre later observed that the same proof gives the irrationality of $\pi^2$). Wall. in his book on continued fractions, claims that the formula is valid "for all $z$".

*Is there a nice way of determining the poles of $\tan z$ from looking at the
right hand side of this expansion?*