The velocities of the two particles after time $t$ are,
$\vec{v}_{1}=\left(u_{1} \cos \theta_{1}\right) \hat{i}+\left(u_{1} \sin \theta_{1}-\mathrm{gt}\right) \hat{j}$
$\vec{v}_{2}=\left(u_{2} \cos \theta_{2}\right) \hat{i}+\left(u_{2} \sin \theta_{2}-\mathrm{gt}\right) \hat{j}$
$\vec{v}_{12}=\vec{v}_{1}-\vec{v}_{2}=\left(u_{1} \cos \theta-u_{2} \cos \theta_{2}\right) \hat{i}+\left(u_{1} \sin \theta_{1}-\mathrm{gt}-u_{2} \sin \theta_{2}+\mathrm{gt}\right) \hat{j}$
which is a constant So the path followed by one, as seen by the other is straight line, making a constant angle with the horizontal.
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