$2\sin x +\tan x > 3x$
$0 < x < \pi/2$
How to prove the above inequality?
2026-05-04 15:54:41.1777910081
How to prove the trigonometric inequality without differentiating?
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You can do this by $AM\ge GM$ and we get
$$\dfrac{\cos x+\cos x+\sec^2 x}{3}\ge(\cos x\cdot\cos x\cdot\sec^2x)^\frac13$$ $$\dfrac{\cos x+\cos x+\sec^2 x}{3}\ge1$$ $$2\cos x+\sec^2 x\ge3$$ $$2\cos x+\sec^2x-3\ge0,x\in(0,\pi/2)$$