<s>[INST] Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$ Prove that 
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$
where $|R_8(x)|\leq \frac{x^8}{8!}$
 [/INST] Hint:   $f(x)=\sum_{i=0}^n\frac{f^ {(i)}(x)x^i}{i! }$
 $\text if$ $$|f^{n+1}(x)|\le M_{n+1}$$then $$|R(x)|\le \frac{M_{n+1}|x|^n}{(n+1)!}$$
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