valid: $$\label{Ineq_16}
\Big{(}\forall t \in \big{[}\mbox{\small $-\sqrt{(n+3)(n+4)},\sqrt{(n+3)(n+4)}$}\,\big{]}\Big{)}
\,\,\, \underline{T}_n(t)\leq \underline{T}_{n+4}(t)\leq \cos t.$$

For the value $t=0$ the inequality in [$($\[Ineq\_16\]$)$]{} turns into equality. For the values $t=\mbox{\small $\pm\sqrt{(n+3)(n+4)}$}$ the equality $\underline{T}_n(t)=\underline{T}_{n+4}(t)$ is true.

Let us notice that for the function $\cos x$ we have following order: $$\label{Ineq_17}
\begin{array}{ll}
\underline{T}_{2}^{\cos ,0}\!(x)\!\leq \!\underline{T}_{6}^{\cos ,0}\!(x)\!\leq \!\underline{T}_{10}^{\cos ,0}\!(x)\!\leq \!\underline{T}_{14}^{\cos ,0}\!(x)
\,\leq\,\ldots\,\leq\,\cos x\,\leq\,\ldots
&  \\[1.5 ex]
\leq \overline{T}_{12}^{\cos ,0}(x)\!\leq\!\overline{T}_{8}^{\cos ,0}(x)\!\leq\!\overline{T}_{4}^{\cos ,0}(x)\!\leq\!\overline{T}_{0}^{\cos ,0}(x)\ \mbox{for} \ x\!\in\!\left[\,0,\sqrt{12}\,\right].
\end{array}$$

Proofs of previous Lemmas given above are presented in the paper [@[9]].

Proofs
======

In order to prove Theorem 2.1. and Theorem 2.2. we will separately observe left and right sides of inequalities.

**The proof of Theorem 2.1.**

Transforming inequality (\[Ineq\_7\]) we have following considerations.

Proving the left side of inequality $$\label{Ineq_18}
2+\left(\nfrac{8}{45}-a(x) \right) x^3 \tan x<\left(\nfrac{\sin ⁡x}{x} \right)^{\!2}+\nfrac{\tan x}{x},$$ for $x \!\in\! \left(0,\nfrac{\pi}{2}\right)$. The inequality (\[Ineq\_18\]) is equivalent to the mixed trigonometric inequality $$\label{Ineq_19}
\begin{array}{lcl}
f(x)
\!\!&\!\!=\!\!&\!\!
1\!-\!8 x^2\!+h_1(x)\cos 4 x + h_2(x)\cos2 x + h_3(x)\sin2 x \\[1.0 ex]
\!\!&\!\!=\!\!&\!\!
1\!-\!8 x^2\!-\!\cos4 x\!-\!8 x^2 \!\cos2 x\!+\!\left(4 x\!-\!4\left(\nfrac{8}{45}\!-\!a(x)\right) x^5 \right)\sin2 x\!>\!0,
\end{array}$$ for $x \!\in\!\left(0,\nfrac{\pi}{2} \right)$, and $h_1(x)\!=\!-1<0$, $h_2(x)\!=\!-8 x^2<0$, $h_3(x)\!=\!4x-4\left(\nfrac{8}{45}-a(x)\right)x^5$.

Now let us consider two cases:

[**(A/I)**]{} [$x\!\in\!(0,1.57]$]{} Let us determine sign of the polynomial $h_3(x)$. As we 