Consider the figure given below : PQRS is a square of side 1 unit and Q, S are the centres of the two circles. The area of the shaded portion is

Place PQRS as square with S=(0,0) and Q=(1,1); side = 1 so distance between centers d = √2. The shaded region is the lens (intersection) of two unit circles. For two equal circles (radius r=1) with center separation d, intersection area = 2·r^2·arccos(d/(2r)) - (d/2)·√(4r^2 - d^2). With d=√2 this gives 2·arccos(√2/2) - (√2/2)·√(2) = 2·(π/4) - 1 = π/2 - 1. Hence the shaded area equals π/2 − 1, which matches option 4. The method follows standard quarter-circle/overlap techniques for square problems of this type [1].

Sources

1. [1] https://www.aimath.org/~circle/theteacherscircle.org/resources/materials/TShubin(Mostly)Simple(MostlyArea)ProblemswithSolutions.pdf