If a line makes angle $\frac{\pi}{3}$ and $\frac{\pi }{4}$ with x-axis and y-axis respectively, then the angle made by the line with z-axis, is

$\frac{\pi}{3}$

We have, $ \alpha = \frac{\pi}{4}, \beta = \frac{\pi}{4}$

$∴ l = cos \frac{\pi}{3}= \frac{1}{2}, m = cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}$

Suppose the line makes angle $γ$ with z-axis and has direction cosines $ l, m, n$. Then,

$l^2 +m^2 + n^2 = 1 $

$⇒\frac{1}{4} +\frac{1}{2} + n^2 = 1⇒ n = ± \frac{1}{2}⇒ cos \gamma = ± \frac{1}{2}⇒ \gamma = \frac{\pi}{3} \, or \, \frac{2\pi}{3}$