I am trying to prove that given $A,B,C,D,E,F \in \big]0,\frac{\pi}{2}\big]$ fixed and $A+C \geq E$ 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?



Inequality involving sine and cosine
I am afraid it is false. Take $F=0$, $A=C$, $E=2A$, $\mu=1/2$. Then we are given $\cos B+\cos D=2\cos A$ and should prove $\cos B/2+\cos D/2\geqslant 2\cos A/2$. But if $\cos B=x$, then $\cos B/2=\sqrt{(1+x)/2}$, this function is concave, thus inverse inequality $\cos B/2+\cos D/2\leqslant 2\cos A/2$ holds.


