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

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

\(x^{\prime}=x_1+x_2\)

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

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

Now as given in question

\(2 a \cos \frac{\phi_1-\phi_2}{2}=a\)

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

\(\frac{\phi_1-\phi_2}{2}=\frac{\pi}{3}\)

\(\phi_1-\phi_2=\frac{2 \pi}{3}\)