$\!s= x^2$ we can notice that the polynomial $P_6(x)$ can be transformed into polynomial of $3^{rd}$ degree $$\label{Ineq_58}
\begin{split}
P_3(s)=(8\pi^6\!-\!672\pi^4\!+\!60480)s^3\!+\!(\!-\!168\pi^6\!+\!15120\pi^2)s^2\!+\!3780\pi^4s\!+\!945\pi^6,
\end{split}$$ $s\!\in\!(0,2.3409]$. Using MATLAB software we can determine the real numerical factorization of the polynomial $$\label{Ineq_59}
\begin{split}
P_3(s)= \alpha(s-s_1)(s^2+ps+q),
\end{split}$$ where $\alpha=2712.204\ldots $, $s_1=-2.221\ldots $, $p=-6.751\ldots $, $q=150.759\ldots $ whereby the inequality $p^2-4q<0$ is true. The polynomial $P_3(s)$ has exactly one real root with a symbolic radical representation and corresponding numerical value $s_1$. Since $P_3(0)>0$ it follows that $P_3(s)>0$ for $s\!\in\!(s_1,\infty)$, so we have following conclusions: $$\label{Ineq_60}
\begin{array}{rccl}
& P_3(s)>0 & \mbox {for} & s\!\in\!(0,2.3409] \subset(s_1,\infty)\\[1.0 ex]
 \Longrightarrow\; & P_6(x)>0 & \mbox {for} & x\!\in\!(0,1.53] \subset \left(0,\nfrac{\pi}{2} \right)\\[1.0 ex]
 \Longrightarrow\; & P_9(x)<0 & \mbox {for} & x\!\in\!(0,1.53] \subset \left(0,\nfrac{\pi}{2} \right).
\end{array}$$ According to the Lemmas 2.3. and 2.4. and description of the method based on (14) and (17), the following inequalities: $\cos y > \underline T_{k}^{\cos ,0}(y)(k=22)$, $\cos y > \underline T_{k}^{\cos ,0}(y)(k=14)$, $\sin y < \overline T_k^{\sin ,0}(y)(k=13)$ are true, for $y\!\in\!\left(0, \sqrt{(k+3)(k+4)}\right)$. For $x\!\in\!(0,1.53]$ it is valid: $$\label{Ineq_61}
\begin{array}{rcl}
f(x)\!\!&\!\!>\!\!&\!\!Q_{22}(x)\!=\!8x^2\!-\!1\!+\!8x^2  \underline T_{14}^{\cos ,0}(2x)\!+\! \underline T_{22}^{\cos ,0}(4x)\!+\!P_9(x) \overline T_{13}^{\sin ,0}(2x),
\end{array}$$ where $Q_{22}(x)$ is the polynomial $$\label{Ineq_62}
\begin{array}{rcl}
Q_{22}(x)\!\!\!&\!\!=\!\!&\!\!\! \left(\nfrac{-33554432}{2143861251406875}\!+\!\nfrac{1024}{5746615875\pi^2}\!-\!\nfrac{4096}{273648375\pi^4}\!+\! \nfrac{8192}{6081075\pi^8} \right)x^{22} \\[1.5 ex]
        \!\!\!&\!\! \!\!&\!\!\! \!+\! \left(\nfrac{140032}{343732764375}\!-\! \nfrac{1024}{147349125\pi^2}\!+\! 