How to find this equation 2sin(Πx/2) = x^2+1/x^2 then the value of x-1/x

Since the range of sine function is [-1,1]. Thus the range of LHS is [-2,2].And applying Arithmetic Mean => Geometric Mean. The range of RHS is from 2 to infinity.Thus the only solution of this equation is when LHS = RHS = 2.Solving LHS = 2, we get solution as x = 4n + 1 where n = 0,1,,2,3,4......but only n = 0 which gives x = 1 satisfies RHS also. Thus, the value of x - 1/x = 0.