Prove that : $\displaystyle \forall x > 0:1 - \frac{{x^2 }}{2} \le \cos x \le 1
$
We have to prove the first inequality. The second is obviously.
Let $\displaystyle f(x)=1-\frac{x^2}{2}-\cos x, \ x>0$.
$\displaystyle f'(x)=-x+\sin x$
$\displaystyle f''(x)=-1+\cos x\leq 0, \forall x\in\mathbb{R}\Rightarrow$ f' is decreasing.
Then, $\displaystyle \forall x>0\Rightarrow f'(x)\leq f'(0)=0\Rightarrow$ f is decreasing.
Then, $\displaystyle \forall x>0\Rightarrow f(x)<\leq f(0)=0\Rightarrow 1-\frac{x^2}{2}\leq \cos x$