Given that, two particles P and Q describes SHM of same amplitude a and frequency f. Let equation of motion of two particles are x1=a‌sin‌ω‌t and x2=a‌sin(ωt+ϕ) where, ϕ be the phase difference. ∴ Path difference, x2−x1=a‌sin(ωt+ϕ)−a‌sin‌ω‌t
x2−x1=2a‌sin(
ωt+ϕ−ωt
2
)‌cos(
ωt+ϕ+ωt
2
)
[Using identity sin‌A−sin‌B=2‌sin(
A−B
2
).cos(
A+B
2
)] ∴ x2−x1=2a‌sin(
Ï•
2
)‌cos(ω+
Ï•
2
) The distance x2−x1 will be maximum when cos(ωt+
Ï•
2
)=1 ∴ Maximum distance x2−x1=√2a =2a‌sin‌