I've read explanations on the internet that the product of two angles is not something fundamentally important. So, the sin of product of two angles cannot be expressed in terms of sin of individual angles. But in calculus, we often come across these functions, we deal with functions involving sin of squares of angles, roots of angles, and even logarithm of angles. So, these functions could be important. And, if a formula doesn't exist, then is there a proof that such a formula of sin(x^2) in terms of sin(x) can't exist?