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I recently asked this question here Inequality involving sine and cosine It turned out that with the conditions I required there, the inequality does not hold. I tried to add extra conditions now, therefore I made a new question:

I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$, $B+D \geq E$, $A+D≥F$ and $B+C≥F$ and the following equation holds for $\mu = 1$:

$$\sin(\mu A)\cos(\mu B) + \sin(\mu C)\cos(\mu D) - \sin(\mu E)\cos(\mu F) \geq 0$$ then it holds for all $\mu\in [0,1]$.

Note obviously if $B \leq F$ and $D \leq F$ the statement is trivial to prove.

I checked numerically and it holds, however whatever I tried to prove it did not work.

Any ideas?

I would like to find a proof of the above. The conditions above should hopefully be sufficient now as far as I can tell.

Edit: As pointed out in the comments, with the four conditions $A+C \geq E$, $B+D \geq E$, $A+D≥F$ and $B+C≥F$ the statement should hold. However I could not prove it yet. With only 3 of the conditions counterexamples can be found.

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    $\begingroup$ $A = 1/6, B = 1/5, C=1/3, D=3/2, E=1/2$ and $F = 19/16$. We have $f(1) > 0$ and $f(1/2) < 0$? $\endgroup$
    – River Li
    Commented Sep 11, 2019 at 10:35
  • $\begingroup$ @River Li if that is true then I also need to require $A+D \geq F$ and $B+C \geq F$ which is unfortunate and will make the whole thing more complicated. But I will try to prove it with those conditions. $\endgroup$
    – Loreno Heer
    Commented Sep 12, 2019 at 7:13
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    $\begingroup$ It seems that any three of these four are not enough. With four, I did not find counterexample. $\endgroup$
    – River Li
    Commented Sep 12, 2019 at 9:55

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