Well, let's define the new quantities $s=\frac{\alpha+\beta}{2}$ and $d=\frac{\alpha-\beta}{2}$. With those quantities the expression reads $E:=\cos(s+d)\cos(s-d)-c\sin(s+d)\sin(s-d).$ The question now is: For which values of $c$ is this expression independent from $d$?
Let's apply the standard addition theorems to get $E =(\sin s\cos d+\cos s\sin d)(\sin s\cos d-\cos s\sin d) -c(\cos s\cos d-\sin s\sin d)(\cos s\cos d+\sin s\sin d).$
Using $(a+b)(a-b) = a^2-b^2$, we therefore get $E=\sin^2 s\,\cos^2 d-\cos^2s\, \sin^2d - c(\cos^2 s\,\cos^2 d-\sin^2 s\,\sin^2 d).$
Now we collect the functions of $s$ to get $E = \sin^2 s(\cos^2 d+c\,\sin^2 d) - \cos^2s(\sin^2 d+c\,\cos^2 d).$
Now it is easy to see that the only possibility that $E$ is independent from $d$ (and therefore in the original form depends only on the sum) is $c=1$, where $\sin^2 d+\cos^2d=1$