$x_1=a \sin \left(\omega t+\phi_1\right)$

$x_2=a \sin \left(\omega t+\phi_2\right)$

$x^{\prime}=x_1+x_2$

$=a\left[\sin \left(\omega t+\phi_1\right)+\sin \left(\omega t+\phi_2\right)\right]$

$=2 a \sin \left(\omega t+\frac{\phi_1+\phi_2}{2}\right) \cos \left(\frac{\phi_1-\phi_2}{2}\right)$

Now as given in question

$2 a \cos \frac{\phi_1-\phi_2}{2}=a$

$\cos \left(\frac{\phi_1-\phi_2}{2}\right)=\frac{1}{2}$

$\frac{\phi_1-\phi_2}{2}=\frac{\pi}{3}$

$\phi_1-\phi_2=\frac{2 \pi}{3}$