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?