Isolation of variables within a trigonometric sum

Question

is it possible to isolate/separate variables within a trigonometric sum? For example, I can't entirely separate f(a,b) = cos(a+b) into something of the form f(a)f(b) or even f(a)+f(b). Can you point out any trigonometric identity that can help with that?

Thanks

Answer 1

Here's why you can't. If there was a way to write $$\cos(a+b) = f(a)f(b),$$ then it would follow that $$\cos(2a) = f(a)f(a) = (f(a))^2.$$ But, for certain ranges of $a$, $\cos(2a)<0$ which contradicts the right hand side of the above equation.

Similarly, if $\cos(a+b) = f(a)+f(b)$, then $\cos(0) = f(a)+f(-a)$. Cosine is an even function, which must mean that $f(a) = f(-a)$. It would follow that $$\cos(0) = 1 = 2f(a) \implies f(a) = 1/2.$$ Therefore, $$\cos(2a) = 2f(a) = 2(1/2) = 1.$$ Because the value of $a$ was arbitrary, this would imply that cosine is a constant function, which is false.

I imagine that you might be working with something more complicated, but you might try applying similar logic as for this simple example.