The direction angles made by a line ' $L$ ’ with the coordinate axes, are $\alpha ,\beta ,\gamma$.  Then the range of ${\sin }^2\alpha +{\sin }^2\beta$

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

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

Hence, ${\sin }^2\alpha +{\sin }^2\beta =2-{\sin }^2\gamma$

The minimum value of ${\sin }^2\gamma$is zero and its maximum value is 1

So that the range of  ${\sin }^2\alpha +{\sin }^2\beta$is $\boxed{\left[1,2\right]}$