The answer depends if you are working on-site or off-site. If you are working directly on the Telligent Evolution site then you can just call the REST APIs via ajax calls directly with one caveat, you have to add the authorization code header to the requests. See this for additional info:
If you're working off-site then you can still using ajax requests if you're using it from a site using the same domain as your Telligent Evolution site: like you are on tools.mydomain.com and your Telligent Evolution site is on community.mydomain.com. If you are using forms auth then you may have to update the cookie to use the wildcard domain so that it is accessible from both sites. The accessing user will have to be logged into the Telligent Evolution site first. You will still have to send the authorization code header so see the doc linked above.
This will provide more examples of authenticating in general: