Given,

A line $L$ is common to the planes $x-y+z+2=0$ and $2x+y-2z+5=0$ then the direction ratio of the line $L$ is given by cross product of normal vector of given plane,

$DR=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& 1\\ 2& 1& -2\end{array}\right|$

$\Rightarrow DR=\hat{i}+4\hat{j}+3\hat{k}$

Now finding direction cosines of the given line we get,

$DC\text{'}s=\frac{\hat{i}}{\sqrt{1^2+4^2+3^2}}+\frac{4\hat{j}}{\sqrt{1^2+4^2+3^2}}+\frac{3\hat{k}}{\sqrt{1^2+4^2+3^2}}$

$\Rightarrow DC\text{'}s=\frac{\hat{i}}{\sqrt{26}}+\frac{4\hat{j}}{\sqrt{26}}+\frac{3\hat{k}}{\sqrt{26}}$

$\Rightarrow DC\text{'}s=\left(\frac{1}{\sqrt{26}},\frac{4}{\sqrt{26}},\frac{3}{\sqrt{26}}\right)$