Proof about power series inverse functions sinx and cosx

[joantica_]

Hi everyone,

How can I proof that the inverse functions of sinx := 1/sinx = cscx and cosx:= 1/cosx = secx, they both have a construction using power series, near 0?, and also calculating their convergence radius after proving that. I thought using Taylor series can be a good idea, but it doesn't convince me. Also if any theorem about functions says some properties to have a construction using power series it would help me.

Any help will be appreciated, thank you. Best regards.

 

[Dr.Peterson]

How can I proof that the inverse functions of sinx := 1/sinx = cscx and cosx:= 1/cosx = secx, ...

Before anyone can answer you, I think we need to clarify what you are talking about. These are not inverse functions! They are reciprocal functions. So, are you asking about the functions you identified, or about inverse functions? It can't be both.

Please show the exact wording of the problem as given to you.

 

[joantica_]

Sorry, I made a mistake, I'm asking about arcsinx and arccosx. I can't rewrite the original message

Before anyone can answer you, I think we need to clarify what you are talking about. These are not inverse functions! They are reciprocal functions. So, are you asking about the functions you identified, or about inverse functions? It can't be both.

Please show the exact wording of the problem as given to you.

 

[Steven G]

Please state the problem.

 

[Deleted member 4993]

Sorry, I made a mistake, I'm asking about arcsinx and arccosx. I can't rewrite the original message

What do you get when you use Taylor's expansion for those functions?

 

[Dr.Peterson]

Sorry, I made a mistake, I'm asking about arcsinx and arccosx. I can't rewrite the original message

Again, we'd like to see the actual problem you are working on (an image could be good) so we can be sure what you have been asked to do. Before proving anything, it is important to have a clear statement of what is to be proved.

In addition, we'd like to know what you have learned that might be relevant; that's part of the reason we ask for whatever work you have done, because it helps us see where you are.

 

[joantica_]

Good news, I have been able to solve the problem. Anyway, there is another question I have, but I don’t know if I should open a new post or not. However, the purpose of it it’s essentially the same but I find impossible to use the same methods I used for arccosx and arcsinx. I’ve attached a photograph. It’s Spanish but I traduce; Proof that the function given can be expressed as a power serie in a 0-environment (I don’t know how to express in English near 0, I hope you’ll understand my words) Also calculate the convergence radius and the first 8 terms of the power serie.
I think that for the convergence radius I can use Leibniz criterion, but I’m now sure as I don’t know how to start proving that the function can be expressed as a power serie, or if there is a ‘trick’ to calculate first of all the first 8 terms.