Let f(x)=x2−2(sin3−sin2)x−(cos3−cos2) then
f(x) is positive ∀x∈R
f(x) assumes both positive and negative values
f(x) = 0 has no real roots
y = f(x) touches the line y = 0
We have
f(x)=x2−2(sin3−sin2)x−(cos3−cos2)
∴ D=4[(sin3−sin2)+(cos3−cos2)]=82sin3−22cosπ4+2+32
As sin3−22>0 and π2<2+32<π
∴ D1<0
Hence, f(x) > 0 ∀x∈R.