1.Using $\displaystyle \varepsilon-\delta$ definition,prove that f(x)=cosx is continuous.
Then if you want to prove it then do it.
I'll do it for you because I want to do it.
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It's easy to prove that $\displaystyle |\sin x-\sin y|\le|x-y|$ for each $\displaystyle x,y\in\mathbb R,$ in particular put $\displaystyle y=0$ and we get this useful inequality $\displaystyle |\sin x|\le|x|.$ As for the sake of the problem given $\displaystyle \epsilon>0,$ then for $\displaystyle 0<\delta=\epsilon,$ we have
$\displaystyle \left| \cos (x)-\cos (a) \right|=\left| 2\sin \left( \frac{x+a}{2} \right)\sin \left( \frac{x-a}{2} \right) \right|\le 2\left| \sin \left( \frac{x-a}{2} \right) \right|,$ hence $\displaystyle |\cos (x)-\cos(a)|=|x-a|<\epsilon.$
As required.