After inelastic collision mass of the combination is $\text{2m}$

$v\text{'}$ be the velocity of the combined mass

Using conservation of momentum along $X$ and $Y$-axis

$2m{v}^{\mathrm{\prime }}\sin \mathrm{\theta }=\frac{mv}{\sqrt{2}}+\frac{mv\sqrt{3}}{2}$

$2m{v}^{\mathrm{\prime }}\cos \mathrm{\theta }=\frac{mv}{2}-\frac{mv}{\sqrt{2}}$

$\frac{2\mathrm{sin\theta }}{2\mathrm{cos\theta }}=\frac{\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}}{\frac{1}{2}-\frac{1}{\sqrt{2}}}=\frac{\sqrt{2}+\sqrt{3}}{1-\sqrt{2}} \mathrm{ }$

$\tan \mathrm{\theta }=\frac{\sqrt{2}+\sqrt{3}\mathrm{ }}{1-\mathrm{ }\sqrt{2}}$