sine and cosine of multiple angles.

Let us mention some facts from [@[7]]. Let $\varphi : [a,b] \longrightarrow \mathbb{R}$ be a function which is differentiable on a segment $[a,b]$ and differentiable arbitrary number of times on a right neighbourhood of the point $x=a$ and denote by $T^{\varphi, a}_{m}(x)$ the Taylor polynomial of the function $\varphi(x)$ in the point $x=a$ of the order $m$. If there is some $\eta\!>\!0$ such that holds: , for $x \!\in\! (a,a+\eta) \!\subset\! [a,b]$; then let us define $\overline{T}^{\,\varphi,a}_{m}(x)=T^{\,\varphi, a}_{m}(x)$ and $\overline{T}^{\,\varphi,a}_{m}(x)$ present an upward approximation of the function $\varphi(x)$ on right neighbourhood $(a,a+\eta)$ of the point $a$ of the oreder $m$. Analogously, if there is some $\eta>0$ such that holds: , for $x \!\in\! (a,a+\eta) \!\subset\! [a,b]$; then let us define $\underline{T}^{\varphi,a}_{\,m}(x)=T^{\,\varphi, a}_{m}(x)$ and $\underline{T}^{\varphi,a}_{\,m}(x)$ present a downward approximation of the function $\varphi(x)$ on right neighbourhood $(a,a+\eta)$ of the point $a$ of the order $m$. Let us note that it is possible to analogously define upward and downward approximations on some left neighbourhood of a point.

According to the paper [@[7]] following Lemmas are true:

[**(i)**]{} For the polynomial $T_n(t)=\!\!\!\!\displaystyle\sum\limits_{i=0}^{(n-1)/2}
\!\!\dfrac{(-1)^it^{2i+1}}{(2i+1)!}$, where $n=4k+1$,\

$k\in \mathbb{N}_{0}$, it is valid: $$\label{Ineq_10}
\Big{(}\forall t \in \big{[}0, \mbox{\small $\sqrt{(n+3)(n+4)}$}\,\big{]}\Big{)}\,\overline{T}_n(t)
