ex]
\!\!&\!\! \!\!&\!\!-\!1451520\pi)x^3+12(240\pi^8-21504\pi^6+1935360\pi^2) x^2\\ [1.0 ex]
\!\!&\!\! \!\!&\!\!+6(-160\pi^9+16128\pi^7-1451520\pi^3) x +120\pi^{10}\\ [1.0 ex]
\!\!&\!\! \!\!&\!\!-14112\pi^8+1330560\pi^4.
\end{array}$$ The polynomial $\hat{P}_6^{''}(x)$ has no real numerical roots for interval $x\!\in\!(0,c_2)$ whereby the function $\hat{P}_6^{''}(x)$ is positive function for $ x\!\in\!(0,c_2)$. That further means that the function $\hat{P}_6^'(x)$ is monotonically increasing function for $ x\!\in\!(0,c_2)$.

The function $\hat{P}_6^{'}(x)$ has root for $x\!=\!\nfrac{\pi}{2}$, also holds that $\hat{P}_6^{'}(c_2)<0$, so we can conclude that $P_6^{'}(x)<0$ for $ x\!\in\!(0,c_2)$ and the function $P_6(x)$ is monotonically decreasing for $ x\!\in\!(0,c_2)$. The function $P_6(x)$ has no roots for $ x\!\in\!(0,c_2)$ and $P_6(c_2)>0$ so we have the following: $$\label{Ineq_73}
\begin{array}{rccl}
&\hat{P}_6(x)>0 & \mbox {for} & x\!\in\!(0,c_2)         \\[1.0 ex]
\Longrightarrow\; & \hat{P}_9(x)<0 & \mbox {for} & x\!\in\!(0,c_2).
\end{array}$$ According to the Lemmas 2.3. and 2.4. and description of the method based on (14) and (17), the following inequalities: $\cos y > \underline  T_{k}^{\cos ,0}(y)(k=2)$ , $\cos y < \overline T_{k}^{\cos ,0}(y)(k=4)$, $\sin y < \overline T_k^{\sin ,0}(y)(k=1)$ are true, for $y\!\in\!\left(0, \sqrt{(k+3)(k+4)}\right)$. For $x\!\in\!(0,c_2)$ it is valid: $$\label{Ineq_74}
\begin{array}{rcl}
\hat{f}(x)\!\!&\!\!=\!\!&\!\!f\!\left(\nfrac{\pi}{2}\!-\!x\right)\!>\hat{Q}_{10}(x)=\!8\left(\nfrac{\pi}{2}\!-\!x\right)^{\!2}\!-\!1\!+\! \underline T_2^{\cos ,0}(4x)\! \\[1.0 ex]
\!\!&\!\! \!\!&\!\! - 8\left(\nfrac{\pi}{2}\!-\!x\right)^{\!2} \overline T_4^{\cos ,0}(2x)+\hat{P}_9(x) \overline T_1^{\sin ,0}(2 x),
\end{array}$$ where $\hat{Q}_{10}(x)$ is the polynomial $$\label{Ineq_75}
\begin{array}{rcl}
\hat{Q}_{10}(x)\!\!&\!\!=\!\!&\!\!\left(-\nfrac{2048}{\pi^8}-\nfrac{256}{945\pi^2}+\nfrac{1024}{45\pi^4}\right) x^{10}+\left(\nfrac{128}{105\pi}-\nfrac{512}{5\pi^3}+\nfrac{9216}{\pi^7}\right) x^9 \\[1.75 ex]
\!\!&\!\! \!\!&\!\!+\left(-\nfrac{64}{27}+\nfrac{1024}{5\pi^2}-\nfrac{18432}{\pi^6}\right) x^8+\left(-\nfrac{3584}{15\pi}+\nfrac{21504}{\pi^5}+\nfrac{352}{135}\pi\right) x^7                       