Any idea for proving the following inequality: $5+8\cos x+4 \cos 2x+ \cos3x\geq 0$ for all real x? I've tried trigonometric identities to make squares appear, and other tricks; but nothing has worked well.
2026-05-16 19:07:39.1778958459
(Elementary) Trigonometric inequality
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Note that $\cos 2x = 2\cos^2 x - 1$ and $\cos 3x = 4\cos^3 x - 3\cos x$.
So, we want to prove that $1+5\cos x + 8\cos^2x + 4\cos^3 x \ge 0$.
The left side is a polynomial in $\cos x$ which can be factored as $(1+\cos x)(1+2\cos x)^2$.
Can you take it from here?