急！！！ 證明sin^2(x)小於等於2(x絕對值) for x屬於R?

: prove that sin^2x≤2｜x｜ for ∀x∈R
Sol
(1) x>=1
2x>=2
Sin^2 x<=1
Sin^2 x<=2
Sin^2 x<=2x
Sin^2 x<=2｜x｜
(2) 1>x>=0
f(x)=2｜x｜－Sin^2 x=2x－Sin^2 x
f’(x)=2－2SinxCosx=2－Sin(2x)>0
f(0)=0－0=0
f(1)=2－Sin^2 1>0
f(x)>=0
2｜x｜>Sin^2 x
(3) 0>=x>=－1
f(x)=2｜x｜－Sin^2 x=－2x－Sin^2 x
f’(x)=－2－2SinxCosx=－2－Sin(2x)<0
f(0)=0－0=0
f(－1)=2－Sin^2 (－1)>0
f(x)>=0
(4) x<=－1
2x<=－2
－2x>=2
Sin^2 x<=1
Sin^2 x<=2
Sin^2 x<=－2x
Sin^2 x<=2｜x｜
2｜x｜>Sin^2 x