exactly two simple real roots with a symbolic radical representation and corresponding numerical values $ x_1$, $ x_2$. Since $P_4(0)>0$ it follows that $P_4(x)>0$ for $ x\!\in\!(x_1, x_2)$, so we have following conclusion: $$\label{Ineq_87}
\begin{array}{rccl}
                  & P_4(x)>0 & \; \mbox {for} \; & x\!\in\!(0,0.98 ] \subset(x_1, x_2) \\[1.0 ex]
\Longrightarrow\; & P_6(x)>0 & \; \mbox {for} \; & x\!\in\!(0,0.98].
\end{array}$$ According to the Lemmas 2.3. and 2.4. and description of the method based on (14) and (17), the following inequalities: $\cos y < \overline T_{k}^{\cos ,0}(y)(k=12)$ , $\cos y < \overline T_{k}^{\cos ,0}(y)(k=8)$, $\sin y > \underline T_k^{\sin ,0}(y)(k=7)$ are true, for $y\!\in\!\left(0, \sqrt{(k+3)(k+4)}\right)$. For $x\!\in\!(0,0.98] $ it is valid: $$\label{Ineq_88}
\begin{array}{rcl}
f(x)\!\!&\!\!>\!\!&\!\!Q_{13}(x)\!=\!1\!-\!8x^2\!-\! \overline T_{12}^{\cos ,0}(4x)\!-\!8 x^2 \overline T_8^{\cos ,0}(2x)\!+\!P_6(x) \underline T_7^{\sin ,0}(2x) ,
\end{array}$$ where $Q_{13}(x)$ is the polynomial $$\label{Ineq_89}
\begin{array}{rcl}
Q_{13}(x)\!\!&\!\!=\!\!&\!\! \left(-\nfrac{1024}{63\pi^5}+\nfrac{512}{315\pi^3}\right) x^{13}+\left(\nfrac{1024}{105\pi^4}-\nfrac{256}{315\pi^2}-\nfrac{16384}{467775}\right) x^{12}\\ [2.5 ex]
\!\!&\!\! \!\!&\!\!+\left(\nfrac{512}{3\pi^5}-\nfrac{256}{15\pi^3}\right) x^{11}+\left(-\nfrac{512}{5\pi^4}+\nfrac{128}{15\pi^2}+\nfrac{3376}{14175}\right) x^{10}\\ [2.5 ex]
\!\!&\!\! \!\!&\!\!+\left(-\nfrac{2560}{3\pi^5}+\nfrac{256}{3\pi^3}\right) x^{9}+\left(\nfrac{512}{\pi^4}-\nfrac{128}{3\pi^2}-\nfrac{64}{63}\right) x^{8}\\ [2.5 ex]
\!\!&\!\! \!\!&\!\!+\left(\nfrac{1280}{\pi^5}-\nfrac{128}{\pi^3}\right) x^{7}+\left(-\nfrac{768}{\pi^4}+\nfrac{64}{\pi^2}+\nfrac{64}{45}\right) x^{6}\\ [2.5 ex]
\!\!&\!\!=\!\!&\!\! -\nfrac{16}{467775}\nfrac{x^6}{\pi^5} Q_7(x),
\end{array}$$ for $ x\!\in\!(0,0.98]$. Then, we have to determine sign of the polynomial $$\label{Ineq_90}
\begin{array}{rcl}
Q_7(x)
\!&\!\!\!=\!\!\!&\!(-47520\pi^2+475200)x^7+(1024\pi^5+23760\pi^3-285120\pi)x^6           \\[1.25 ex]
\!&\!\!\! \!\!\!&\!+(498960\pi^2-4989600)x^5+(-6963\pi^5-249480\pi^3                     \\[1.25 ex]
\!&\!\!\! \!\!\!&\!+2993760\pi)x^4+(\!-\!2494800\pi^2\!+\!24948000)x^3\!+\!(29700\pi^5\! \\[1.25 ex]
\!&\!\!\! \!\!\!&\!+1247400\pi^3\!-\!14968800\pi)x^2+(3742200\pi^2-37422000)x            \\[1.25 ex]
\!&\!\!\! \!\!\!&\!-41580\pi^5-1871100\pi^3+22453200\pi,
\end{array}$$ for $ x\!\in\!(0,0.98]$, which is the polynomial 