 Solution

At the central values, $z=\sqrt{\tan 45^{\circ }}+0.10^2=1.010$. For independent input standard errors, linear propagation givesThe angular derivative here is with respect to radians, so ${\sigma }_x=2\pi /180=0.03491$. At $x=\pi /4$, the derivative is one, andThus $\boxed{z=1.010\pm 0.035}$, with the stated measurement uncertainty interpreted as one standard error. Correlated inputs would require a covariance term; their correlation is not supplied.