Two particles P and Q describe S.H.M. of same amplitude a, same...

\mathrm{x}_{1}=\mathrm{a} \sin \left(\omega \mathrm{t}+\phi_{1}\right), \mathrm{x}_{2}=\mathrm{a} \sin \left(\omega \mathrm{t}+\phi_{2}\right)\Rightarrow \quad\left|\mathrm{x}_{1}-\mathrm{x}_{2}\right|=2 \mathrm{a} \sin \left(\omega \mathrm{t}+\frac{\phi_{1}+\phi_{2}}{2}\right) \cos \left(\frac{\phi_{1}-\phi_{2}}{2}\right) Tomaximize \left|\mathrm{x}_{1}-\mathrm{x}_{2}\right|$: \sin \left(\omega \mathrm{t}+\frac{\phi_{1}+\phi_{2}}{2}\right)=$1\Rightarrow $\mathrm{a} \sqrt{2}=$2 $\mathrm{a} \times$1 \times \cos \left(\frac{\phi_{1}-\phi_{2}}{2}\right)\Rightarrow \frac{1}{\sqrt{2}}=\cos \left(\frac{\phi_{1}-\phi_{2}}{2}\right) \Rightarrow \frac{\pi}{4}=\frac{\phi_{1}-\phi_{2}}{2}\Rightarrow \phi_{1}-\phi_{2}=\frac{\pi}{2}$$