If a line makes angles of $90^\circ, 135^\circ$ and $45^\circ$ with the $x, y$ and $z$ axes, respectively, then its direction cosines are:

$0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$

The correct answer is Option (1) → $0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ ##

Direction cosines of a line making angle $\alpha$ with X-axis, $\beta$ with Y-axis and $\gamma$ with Z-axis are $l, m, n$

$l = \cos \alpha, m = \cos \beta, n = \cos \gamma$

Here, $\alpha = 90^\circ, \beta = 135^\circ, \gamma = 45^\circ$

So, direction cosines are

$l = \cos 90^\circ = 0$

$m = \cos 135^\circ = \cos(90^\circ + 45^\circ)$

$= -\sin 45^\circ = -\frac{1}{\sqrt{2}}$

and $n = \cos 45^\circ = \frac{1}{\sqrt{2}}$

Therefore, direction cosines are $0, -\frac{1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$.