Let sin x and cos x be defined by their power series expansions
how can we use term by term differentiation to show that
sin x = cos x and
cos x = − sin x.
Exactly the point!
Let sin x and cos x be defined by their power series expansions …
So IF you were to define the sine & cosine functions by what we know their power series are. Guess what!! No surprise that
It's not so much circular reasoning as it is an exercise to help students verify that the power series have the same properties as the functions they represent - indeed they are identical.