A straight line which makes an angle of 60° with each of y and z axes, includes with x-axis at an angle

45°

Suppose the line makes angle $\gamma $ with z-axis. Then, its direction cosines are

$l = cos 60° = \frac{1}{2}, m = cos 60 ° = \frac{1}{2}, n = cos \gamma $

$∴ l^2 + m^2 + n^2 = 1 ⇒ \frac{1}{4}+\frac{1}{4}+cos^2 \gamma = 1 $

$⇒ cos^2 \gamma = \frac{1}{2}⇒ cos \gamma = ± \frac{1}{\sqrt{2}} ⇒ \gamma = 45° \, or, 135°$