recaptcha https problem on https://api-secure.recaptcha.net/
As Remy says, you need to update the links to point at Google's servers. See the announcement regarding the expiration of the old URL.
If you are using the .NET Library then you can update to the latest version
If you did not use the library then you can see the modifications made to the library here, and update your URLs in the same manner.
i.e. update https://api-secure.recaptcha.net to https://www.google.com/recaptcha/api
ReCaptcha was acquired by Google. Have you updated your code to use the new Google links yet?