If $\alpha, 2 \alpha, 3 \alpha$ are angles made by a ray with $\mathrm{OX}, \mathrm{OY}, \mathrm{OZ}$ axes…
If $\alpha, 2 \alpha, 3 \alpha$ are angles made by a ray with $\mathrm{OX}, \mathrm{OY}, \mathrm{OZ}$ axes respectively then all the possible values of $\alpha$ are
$\frac{\pi}{6}, \frac{\pi}{12}$
$\frac{\pi}{6}, \frac{\pi}{3}$
$\frac{\pi}{4}, \frac{\pi}{3}$
$\frac{\pi}{6}, \frac{\pi}{4}$
Solution
If any ray makes $\alpha, \beta$, angles with axes then $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2=1$
$\cos ^2 \alpha+\cos ^2 2 \alpha+\cos ^2 3 \alpha=1$
at $\alpha=\frac{\pi}{4}$
$\cos ^2 \frac{\pi}{4}+\cos ^2 \frac{\pi}{2}+\cos ^2 \frac{3 \pi}{4}=\frac{1}{2}+0+\frac{1}{2}=1$
$\alpha=\frac{\pi}{6}$
$\cos ^2 \frac{\pi}{6}+\cos ^2 \frac{\pi}{3}+\cos ^2 \frac{\pi}{2}=\frac{3}{4}+\frac{1}{4}=1$
possible values of $\alpha$ are $\frac{\neq}{6}, \frac{\neq}{4}$