The equation
2cos2x2sin2x=x2+x−2;0<x≤π2
has
The given equation is
2cos2(x2)sin2x=x2+1x2
where 0<x≤π2
L.H.S.=2cos2(x2)sin2x=(1+cosx)sin2x∵1+cosx<2 and sin2x≤1 for 0<x≤π2∴(1+cosx)sin2x<2
and R.H.S.=x2+1x2≥2
Therefore, for 0<x≤π2, the given equation is not possible for any real value of x.