is monotonically increasing for $ x\!\in\!(0,c_2)$. Further, $\hat{Q}_8^{'''}(c_2)<0$ and the function $\hat{Q}_8^{'''}(x)$ has the first positive root $ x=1.00733 \ldots $ which follows that $\hat{Q}_8^{'''}(x)<0$ for $ x\!\in\!(0, c_2) \subset(0,1.00733 \ldots)$ and the function $\hat{Q}_8^{''}(x)$ is monotonically decreasing function for $ x\!\in\!(0,c_2)$. $\hat{Q}_8^{''}(c_2)>0$ and the function $\hat{Q}_8^{''}(x)$ has the first positive root $ x=0.45455 \ldots $ which follows that $\hat{Q}_8^{''}(x)>0$ for $ x\!\in\!(0,c_2) \subset(0,0.45455 \ldots)$ and the function $\hat{Q}_8^{'}(x)$ is monotonically increasing function for $ x\!\in\!(0,c_2)$. $\hat{Q}_8^{'}(0)>0$ and the function $\hat{Q}_8^{'}(x)$ has the first positive root $ x=1.16834 \ldots $ which follows that $\hat{Q}_8^{'}(x)>0$ for $ x\!\in\!(0,c_2) \subset(0,1.16834 \ldots)$ and the function $\hat{Q}_8(x)$ is monotonically increasing function for $ x\!\in\!(0, c_2)$. Since we have that $\hat{Q}_8(c_2)<0$ and the function $\hat{Q}_8(x)$ has the first positive root $ x=0.04383 \ldots $ we can conclude following: $$\label{Ineq_79}
\begin{array}{rccl}
& \hat{Q}_8(x)<0 & \mbox {for} & x\!\in\!(0, c_2)\subset(0,0.04383 \ldots)                    \\[1.0 ex]
\Longrightarrow\; & \hat{Q}_{10}(x)>0 & \mbox {for} & x\!\in\!(0,c_2)                         \\[1.0 ex]
\Longrightarrow\; & \hat{f}(x)=f\left(\nfrac{\pi}{2}- x \right)>0 & \rm for & x\!\in\!(0,c_2) \\[1.0 ex]
\Longrightarrow\; & f(x)>0 & \mbox {for} & x\!\in\!\left(1.53,\nfrac{\pi}{2}\right).
\end{array}$$ Hence we proved that the function $f(x)$ is positive on interval $ x\!\in\!(0,1.53]$ we conclude that the function $f(x)$ is positive on whole 