It is rare that giving a recorded statement to the other side's insurance company would help your claim. In our experience, the insurance adjustor you would be talking to is trying to support a bizzare story from their insured driver that makes you at fault.
If you are talking to an adjustor about anything other than co ordinating someone to look at your car, you are probably making things harder for yourself.
Even though you have done nothing wrong, and just want the truth to come out, giving a recorded statement to someone as motivated as an adjustor is a poor way to do that. The best way for the truth to come out is when you have had the help of an experienced trial lawyer. We can help find the evidence that reveals the real truth of an incident and your injuries. Once you get a lawyer, adjustors usually don't even bother to ask for a recorded statement.