If $\alpha ,\beta ,\gamma$ are the angles made by a line ‘$L$’ with the coordinate axes in the positive direction then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

Given $\alpha ,\beta ,\gamma =\frac{\pi }{2}-\beta$

It implies that ${\cos }^2\alpha +{\cos }^2\beta +{\sin }^2\beta =1\Rightarrow {\cos }^2\alpha =0$

Consider the expression 

$\begin{array}{c}{\left(\cos \alpha +\cos \beta +\cos \gamma \right)}^2={\left(0+\cos \beta +\sin \beta \right)}^2\\ ={\cos }^2\beta +{\sin }^2\beta +2\cos \beta \sin \beta \\ =1+\sin 2\beta \end{array}$

Therefore, the required expression is$\boxed{1+\sin 2\beta }$