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.

- Feb 17th 2011, 05:58 AMmaximus101Power series expansion proofs for the differential of sinx and cosx
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. - Feb 17th 2011, 06:35 AMchisigma
- Feb 17th 2011, 07:24 AMProve It
- Feb 17th 2011, 09:21 AMKrizalid
No it's not circular.

Note that he differentiate the power series (given hypothesis) and identified that series as cosine. - Feb 17th 2011, 09:28 AMProve It
- Feb 17th 2011, 09:48 AMSammyS
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.