Given :

$\alpha =90^{\circ},\:\:\beta=135^{\circ}\:\:and\:\:\gamma=45^{\circ}$

The direction cosines are

$\cos \alpha =\cos 90^{\circ}=0$

$\cos \beta =\cos 135^{\circ}=\cos(180^{\circ}-45^{\circ})$

$\qquad\qquad\quad\quad\;\;\;=\large\frac{-1}{\sqrt 2}$

$\cos \gamma =\cos 45^{\circ}=1$

Hence the direction cosines are $ (0,\large\frac{-1}{\sqrt 2}$$,1)$