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I am trying to prove the inequality $$ \begin{aligned} \cos((b - a) t) - \cos((b + a) t) + t \sqrt{n (n - 1)} (a - b)\sin((a + b) t)\\ - t \sqrt{n (n - 1)} (a + b) \sin((b - a) t)\le0 \end{aligned} $$ where $a=\sqrt{k/n}$ and $b=\sqrt{k/(n-1)}$ for $t\in[0,1]$, $k>0$ and $n\ge2$.

After plotting this function, I am convinced that the inequality holds. I tried to prove it using derivative with respect to $t$, but ended up with sines and cosines with coefficients getting more and more complicated. Is there a way to prove this?

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$$\cos\left((b-a)t\right)=\cos(bt-at)=\cos(bt)\cos(at)+\sin(bt)\sin(at)$$ $$\cos((b+a)t)=\cos(bt)\cos(at)-\sin(bt)\sin(at)$$ $$\cos\left((b-a)t\right)-\cos((b+a)t)=2\sin(bt)\sin(at)$$ try from here

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  • $\begingroup$ This is what I use to get the inequality..... $\endgroup$ – Y.Guo Sep 19 '18 at 14:57
  • $\begingroup$ Then what is your question? $\endgroup$ – Henry Lee Sep 19 '18 at 14:58
  • $\begingroup$ Well...I mean I reduce another inequality (which I was trying to prove) to the one presented in the problem using this formula you give. $\endgroup$ – Y.Guo Sep 19 '18 at 15:01
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    $\begingroup$ can you just tell actual question? $\endgroup$ – Narendra Sep 19 '18 at 17:24

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