So I got a problem wrong on my homework because I simplified $\cos (x) \times \cos (y)$ to be $\cos (xy)$, and my teacher wrote on my paper that it can't be simplified. I was wondering if anyone could give me an explanation as to why you can't simplify those two.
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$\begingroup$It is depends what you mean by saying simplify... Sometimes it can be useful to use this identity: $\cos(x) \cos(y) = \frac{1}{2} \left[ \cos(x+y) + \cos(x-y)\right]$ Usually you can change trigonometric function multiplication to sum or backwards.
Multiplication of any two functions with different variables cannot be simplified into one function of one variable. This kind of simplification is extremely rare:
(1) $f(x)f(y)=g(xy)$
Examples of function that fulfill (1) is $f(x)=ax$ or $f(x)=ax^n$. Other function that fulfill (1) probably esoteric...
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