If the pair of lines represented by the combined equation $2{\mathrm{x}}^2+4\mathrm{xy}+{\mathrm{y}}^2=0$ makes angles ${\mathrm{\theta }}_1,{\mathrm{\theta }}_2$with $\mathrm{x}-$axis then $\cos \left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)=$

If ${\mathrm{\theta }}_1,{\mathrm{\theta }}_2$ are the inclination of the lines represented by the combined equation $2{\mathrm{x}}^2+4\mathrm{xy}+{\mathrm{y}}^2=0$ then ${\mathrm{\theta }}_1-{\mathrm{\theta }}_2$ is the angle between those two lines. 
If $\mathrm{\theta }$ is acute angle between pair of lines represented by the equation ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2=0$ then $\mathrm{cos\theta }=\frac{\left|\mathrm{a}+\mathrm{b}\right|}{\sqrt{{(\mathrm{a}-\mathrm{b})}^2+{\left(2\mathrm{h}\right)}^2}}$
For the equation $2{\mathrm{x}}^2+4\mathrm{xy}+{\mathrm{y}}^2=0,\mathrm{a}=2,\mathrm{h}=2,\mathrm{b}=1$ ,   
Hence, 
$\begin{array}{c}\cos \left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)=\frac{3}{\sqrt{1+16}}\\ =\frac{3}{\sqrt{17}}\end{array}$
Therefore, the required value is $\cos \left({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\right)=\boxed{\frac{3}{\sqrt{17}}}$