# Math Help - Prove that

1. ## Prove that

Prove that : $\forall x > 0:1 - \frac{{x^2 }}{2} \le \cos x \le 1

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2. We have to prove the first inequality. The second is obviously.

Let $f(x)=1-\frac{x^2}{2}-\cos x, \ x>0$.

$f'(x)=-x+\sin x$

$f''(x)=-1+\cos x\leq 0, \forall x\in\mathbb{R}\Rightarrow$ f' is decreasing.

Then, $\forall x>0\Rightarrow f'(x)\leq f'(0)=0\Rightarrow$ f is decreasing.

Then, $\forall x>0\Rightarrow f(x)<\leq f(0)=0\Rightarrow 1-\frac{x^2}{2}\leq \cos x$

3. Originally Posted by dhiab
Prove that : $\forall x > 0:1 - \frac{{x^2 }}{2} \le \cos x \le 1

$
This is equivalent to proving

$|x|\geq |\sin x | \geq 0$