Inequality of Huygens

The inequality of Huygens is a mathematical result establishing that, on the interval $\backslash \; left\; of$ \backslash \; mathbb\; \{R\}$,\; the\; following\; inequality\; is\; checked:$ 2\; \backslash \; sin\; X\; +\; \backslash \; tan\; X\; \backslash \; geq\; 3x$.$

Demonstration

That is to say F function which with any X of $\backslash \; left\; associates$ F\; (X)\; =\; 2\; \backslash \; times\; \backslash \; sin\; \{X\}\; +\; \backslash \; tan\; \{X\}\; -\; 3x$.$
Like summons functions of C_∞ class on the interval considered, F is continuously derivable an infinity of time on this same interval.
The function derived from F is the function f' which with any X of the interval considered associates: $f\text{'}\; (X)\; =\; \backslash \; frac\; \{2\; \backslash \; cos^\; \{3\}\; x-3\; \backslash \; cos^\; \{2\}\; x+1\}\; \{\backslash \; cos^\; \{2\}\; \{X\}\}$.
On the interval considered, the following inequalities are checked: $\backslash \; forall\; X,\; \backslash \; cos^\; \{2\}\; X\; >\; 0$ and $\backslash \; forall\; X\; (\backslash \; cos\; X\; -1)\; ^\; \{2\}\; (2\; \backslash \; cos\; \{X\}\; +1)\; \backslash \; geq\; 0$.
Therefore, $\backslash \; forall\; X\; \backslash \; in\; \backslash \; left\; f\text{'}\; (X)\; \backslash \; geq\; 0$ and the function F is increasing on the interval considered.
The equality is noticed: $f\; (0)\; =\; 0$ thus, consequently: $\backslash \; forall\; X\; \backslash \; in\; \backslash \; left\; F\; (X)\; \backslash \; geq\; 0\; \backslash \; Leftrightarrow\; \backslash \; forall\; X\; \backslash \; in\; \backslash \; left\; [\{0\};\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right\; [,\; 2\; \backslash \; sin\; X\; +\; \backslash \; tan\; X\; \backslash \; geq\; 3x$

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