I'm sorry to ask this question , but should ask it may help me to know more
about series theory , It is well known that $\cos x $ and $\sin x$ are
represented by alternative series which hard for me to show wether sinx and
cosx are in this range $[-1,1]$ .
My question here :Is there analytical proof show that $\sin x$ and $\cos x$ both are really in $[-1,1]$ using series theory if it is possible
Note: I do not want to use geometric interpretation because it is standard at all
Thank you for any help