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Question Number 100870 by bramlex last updated on 29/Jun/20

Commented by bemath last updated on 29/Jun/20

$$\pi -\mathrm{e}=0.4233$$$${(\pi -\mathrm{e})}^{\ln (1-2\cos {}^2\mathrm{x})}⩾{(\pi -\mathrm{e})}^0$$$$\mathrm{case}\left(1\right)\Rightarrow \ln (1-2(\frac{1}{2}+\frac{1}{2}\cos 2\mathrm{x}))⩽0$$$$\ln (-\cos 2\mathrm{x})⩽0\Rightarrow -\cos 2\mathrm{x}⩽1$$$$\cos 2\mathrm{x}⩾-1,\mathrm{\forall }\mathrm{x}\in \mathbb{R}$$$$\mathrm{case}\left(2\right)\Rightarrow -\cos 2\mathrm{x}>0;\cos 2\mathrm{x}<0$$$$\frac{\pi }{4}<\mathrm{x}<\frac{3\pi }{4}\Rightarrow x\in (\frac{\pi }{4},\frac{3\pi }{4})★$$

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