\ldots >1.53$ (and $x^*< \nfrac{\pi}{2}$) which is the first positive root of the polynomial $Q_{22}(x)$ defined at (\[Ineq\_62\]).

[**(B/II)**]{} [$ x\!\in\! \left(1.53,\nfrac{\pi}{2} \right)$]{} Let us define the function $$\label{Ineq_68}
\begin{array}{rcl}
\hat{f}(x)\!\!\!&\!\!\!=\!\!\!&\!\!\!f \left(\nfrac{\pi}{2}-x\right)=8\left(\nfrac{\pi}{2}-x\right)^{\!2}\!-\!1\!+\!\hat{h}_1(x)\cos4 x\!+\!\hat{h}_2(x)\cos2 x\!+\!\hat{h}_3(x)\sin2 x \\[1.5 ex]
\!\!\!&\!\!\!=\!\!\!&\!\!\!8\left(\nfrac{\pi}{2}-x\right)^{\!2}-1+\cos4 x-8\left(\nfrac{\pi}{2}-x\right)^{\!2} \cos2 x                                                                  \\[1.5 ex]
\!\!\!&\!\!\! \!\!\!&\!\!\!+\left(4\left(\nfrac{8}{45}-b_1\left(\nfrac{\pi}{2}-x\right) \right)\!\left(\nfrac{\pi}{2}- x\right)^{\!5}-4 \left(\nfrac{\pi}{2}- x \right)\right)\sin2 x,
\end{array}$$ where $x\!\in\!(0,c_2)$ for $c_2=\nfrac{\pi}{2}-1.53=\nfrac{\pi}{2}-\nfrac{153}{100}\left(= 0.04079\ldots\right)$, and $\hat{h}_1(x)=1>0$, $\hat{h}_2(x)=8\left(\nfrac{\pi}{2}\!-\!x\right)^{\!2}>0$, $\hat{h}_3(x)=4\left(\nfrac{8}{45}\!-\!b_1 \left(\nfrac{\pi}{2}\!-\!x\right)\right)\!\left(\nfrac{\pi}{2}\!-\!x\right)^5\!-\!4\left(\nfrac{\pi}{2}\!-\!x\right)$.

We are proving that the function $\hat{f}(x)>0$.

Again, it is important to find sign of the polynomial $\hat{h}_3(x)$. As we see, that polynomial is the polynomial of $9^{th}$ degree $$\label{Ineq_69}
\begin{array}{rcl}
\hat{h}_3(x)
\!\!&\!=\!&\!\!\hat{P}_9(x)=4 \left(\nfrac{8}{45}-b_1 \left(\nfrac{\pi}{2}- x \right)\right)\!\left(\nfrac{\pi}{2}- x \right)^{\!5}-4 \left(\nfrac{\pi}{2}- x \right)                                                         \\[1.5 ex]
\!\!&\!=\!&\!\!\nfrac {32}{45} \left(\nfrac{\pi}{2}- x \right)^{\!5}-\nfrac{32}{945} \left(\nfrac{\pi}{2}- x \right)^{\!7}+ \nfrac{4\mbox{\boldmath $a$}}{14175} \left(\nfrac{\pi}{2}- x \right)^{\!9}-4 \left(\nfrac{\pi}{2}- x \right).
\end{array}$$ Let us determine the sign of the polynomial $$\label{Ineq_70}
\begin{array}{rcl}
\hat{P}_9(x)\!\!&\!=\!&\!\!-\nfrac{1}{945\pi^8}\Big(x(\pi-x)(\pi-2x)(\pi^{12}-12\pi^{11}x+60\pi^{10}x^2-160 \pi^9 x^3 \\[1.25 ex]
\!\!&\! \!&\!\!+240\pi^8x^4\!-\!192\pi^7x^5\!+\!64\pi^6x^6\!-\!168\pi^{10}\!+\!1680\pi^9x\!-\!7056\pi^8x^2            \\[1.25 ex]
\!\!&\! \!&\!\!+16128\pi^7x^3\!-\!21504\pi^6x^4\!+\!16128\pi^5x^5\!-\!5376\pi^4x^6\!+\!30240\pi^6                     \\[1.25 ex]
\!\!&\! \!&\!\!-181440\pi^5x\!+\!665280\pi^4x^2\!-\!1451520\pi^3\!+\!1935360\pi^2x^4                                  \\[1.25 ex]
\!\!&\! \!&\!\!-1451520\pi x^5 +483840x^6)\Big)\!                                                                     \\[1.25 ex]
\!\!&\!=\!&\!\! \!-\!\nfrac{1}{945\pi^8}x(\pi\!-\!2x)(\pi\!-\!x)\hat{P}_6(x),
\end{array}$$ for $ x\!\in\!(0,c_2)$ where $$\label{Ineq_71}
\begin{array} {rcl}
\hat{P}_6(x)\!\!&\!\!=\!\!&\!\!(64\pi^6\!-\!5376\pi^4\!+\!483840)x^6\!+\!(\!-\!192\pi^7\!+\!16128\pi^5\!-\!1451520\pi)x^5 \\[1.25 ex]
\!\!&\!\! \!\!&\!\!+(240\pi^8-2150\pi^6+1935360\pi^2)x ^4+(-160\pi^9+16128\pi^7                                           \\[1.25 ex]
\!\!&\!\! \!\!&\!\!-1451520\pi^3)x ^3+(60\pi^{10}-7056\pi^8+665280\pi^4)x ^2+(-12\pi^{11}                                 \\[1.25 ex]
\!\!&\!\! \!\!&\!\!-1680\pi^9-181440\pi^5)x+\pi^{12}-168\pi^{10}+30240\pi^6.
\end{array}$$ The second derivate of the polynomial $\hat{P}_6(x)$ is the polynomial of $4^{th}$ degree $$\label{Ineq_72}
\begin{array}{rcl}
\hat{P}_6^{''}(x)\!\!&\!\!=\!\!&\!\!30(64\pi^6\!-\!5376\pi^4\!+\!483840)x^4\!+\!20(\!-\!192\pi^7\!+\!16128\pi^5\!\\ [1.0 