Simple answer, "No". You should NEVER give a recorded statement to the insurance company. They will use the statement for their benefit and not yours.
You will receieve a call from the insurance company within a few days from your accident. They will be recording this and all other conversations with you. They will ask how you are feeling and attempt to get your version of the accident. The call will then be transcribed and used later throughout your claims process. You and I are both aware of how something written down may not accurately reflect what occured during the conversation, including your inflection and why you stated what you did. Keep this in mind when responding to questions from the adjustors.
If you have any additional questions after watching our video, please contact our office, we will be happy to assist you.