$\boldsymbol{x}_{2}=20 \sin (\omega t+\phi)$

distance between them is given by

$\left|\boldsymbol{x}_{2}-\boldsymbol{x}_{1}\right|=|20 \sin (\omega t+\phi)-20 \sin \omega t|=20|(\sin (\omega t+\phi)-\sin \omega t)|$

$\Rightarrow\left|\boldsymbol{x}_{2}-\boldsymbol{x}_{1}\right|=40\left|\cos \left(\omega t+\frac{\theta}{2}\right) \sin \left(\frac{\theta}{2}\right)\right|$

The distance between them is maximum when $\cos \left(\omega t+\frac{\theta}{2}\right)=1$ i.e.

$40 \sin \left(\frac{\theta}{2}\right)=20$

$\Rightarrow \sin \left(\frac{\theta}{2}\right)=\frac{1}{2}$

$\Rightarrow \frac{\theta}{2}=\frac{\pi}{\pi}$

$\Rightarrow \theta=\frac{\pi}{3}$