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Show that for all $x\in\mathbb{R},$

\begin{equation} \cos^2\left(x\right)\left(2+\cos 2x\right)-2 \cos^2\left(x+\frac{\sin 2x}{4}\right)\ge0 \end{equation}

I have checked this numerically and this is correct but I cannot show analytically. I have used the following useful formulas but had no luck: \begin{gather} \sin(2x)=2\sin (x)\cos (x),\newline \cos (2x)=2\cos^2(x)-1,\newline \sin^2(x)=1-\cos^2(x). \end{gather} I guess there is some bounding trick that I am missing. Any help is appreciated

JamesV
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1 Answers1

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If you expand the first term on the LHS, and add the second term on the LHS to both sides, you'll get

$2\cos^2(x) + \cos^2(x)\cos(2x) \geq 2\cos^2(x+\frac{\sin(2x)}{4})$

Now, using the identity $\cos(2x)+1 = 2\cos^2(x)$, you get:

$\cos(2x)+1 + \frac{\cos(2x)+1}{2}\cos(2x) \geq \cos(2x+\frac{\sin(2x)}{2})+1$

Subtract $1$ from both sides, and write the LHS by taking the expression into the paranthesis of $\cos(2x)$, you get:

$\cos(2x)(\frac{\cos(2x)+3}{2}) \geq \cos(2x + \frac{\sin(2x)}{2})$

Substitute $y=2x$ at this point, and it is easy to see that for each $y \in \mathbb{R}$ that satisfies the equation, $y + 2\pi k$ and $-y$ also satisfies the equation for $k \in \mathbb{Z}$. Thus, it is enough to consider the inequality for $y \in [0,\pi]$.

$\frac{\cos(y)+3}{2} \geq 1$ for all $y$; thus, if $\cos(y) \geq \cos(y + \frac{\sin(y)}{2})$ for all $y$, the inequality holds.

It can be seen that the equality $\cos(y) = \cos(y + \frac{\sin(y)}{2})$ holds when either $\frac{\sin(y)}{2}=0$ or $2\pi- y = y + \frac{\sin(y)}{2}$

The first case holds when $y\in \{0,\pi\}$ and by simple manipulation and checking the derivatives, the only solution for the second case is $y = \pi$ (of course for $y \in [0,\pi]$).

Thus, as there is no zero for $0\leq y \leq \pi$, either:

$\forall y; \cos(y) \geq \cos(y + \frac{\sin(y)}{2})$

or

$\forall y; \cos(y) \leq \cos(y + \frac{\sin(y)}{2})$

For $y = \pi/2, \cos(\pi/2) > \cos(\pi/2 + \frac{1}{2\sqrt{2}}) \Rightarrow \forall y; \cos(y) \geq \cos(y + \frac{\sin(y)}{2})$

Hence, $\forall y\in \mathbb{R}$;

$\cos(2x)(\frac{\cos(2x)+3}{2}) \geq \cos(2x + \frac{\sin(2x)}{2})$

$\blacksquare$

by24
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