was applied in papers [@[8]] and [@[9]] on inequalities which are closely related.

The Main Results
================

The main purpose of our paper is to extend the intervals defined in theorems given by [C. Mortici]{} [@[6]]. More precisely, we extend the domains $\left(0, 1\right)$ and $\left(\nfrac{\pi}{2}-\nfrac{1}{2}, \nfrac{\pi}{2} \right)$ from the previous theorems to $\left(0, \nfrac{\pi}{2} \right)$. We give the next two statements.

For every $x \!\in\! \left(0, \nfrac{\pi}{2} \right)$ the following inequalities are true$:$ $$\label{Ineq_7}
2+\left(\nfrac{8}{45}-a\left(x \right) \right) x^3 \tan x
<
\left(\nfrac{\sin ⁡x}{x} \right)^{\!2}+\nfrac{\tan x}{x}
<
2+\left(\nfrac{8}{45}-b_{1} \left(x \right)\right) x^3 \tan x,$$ where $a(x)= \nfrac{8}{945} x^2$, $b_{1}(x)= \nfrac{8}{945} x^2- \nfrac{\mbox{\boldmath $a$} }{14175} x^4$ and $
\mbox{\boldmath $a$}
=
\frac{480\pi^6-40320\pi^4+3628800}{\pi^8}
=
17.15041 \ldots \, .
$

For every $x \!\in\! \left(0, \nfrac{\pi}{2} \right)$ the following inequalities are true$:$ $$\label{Ineq_8}
2+\left(\nfrac{16}{\pi^4} +c\left(x\right) \right) x^3 \tan x
<
\left(\nfrac{\sin x}{x}\right)^{\!2}+\nfrac{\tan x}{ x}
<
2+\left(\nfrac{16}{\pi^4}+d\left(x\right) \right) x^3 \tan x,$$ where\

$$c(x)
\!=\!
\left(\nfrac{160}{\pi^5}\!-\!\nfrac{16}{\pi^3}\right)\!\left(\nfrac{\pi}{2}\!-\!x\right),
~
d(x)
\!=\!
\left(\nfrac{160}{\pi^5}\!-\!\nfrac{16}{\pi^3}\right)\!\left(\nfrac{\pi}{2}\!-\!x\right)+\left(\nfrac{960}{\pi^6}-\nfrac{96}{\pi^4}\right)\!\left(\nfrac{\pi}{2}\!-\!x\right)^{\!2}.$$

In [@[7]] is considered a method of proving trigonometric inequalities for mixed trigonometric polynomials: $$\label{Ineq_9}
f(x)=\sum\limits_{i=1}^{n}\alpha_{i}x^{p_{i}}\!\cos^{q_{i}}\! x\sin^{r_{i}}\! x>0,$$ for $x \!\in\!(\delta_{2},\delta_{1})$, $\delta_{2}\!<\!0\!<\!\delta_{1}$, where $\alpha_{i}
\!\in\! \mathbb{R} \!\setminus\! \lbrace0\rbrace$, and . One method of proving inequalities in form (\[Ineq\_9\]) is based on transformation, using the sum of 