You need to look at this issue from another perspective - the mutual trust.
There is no way to prove the number of hours you bill your client is the actual number of hours you spend on the task you are assigned.
Before you take the task, you should ask the client how many hours they expect you to spend for that task. Then you tell them how many hours you will need. You and your client eventually reach a mutually acceptable number of hours needed before you start the task.
You go do that task, document the hours you spend in a readable format. After you finish the task, report the result back to the client and then bill them with the invoice and the time sheet.
If the number of hours is within the reasonable range, say you told them you need 40 hours to do the task and you put in 42 hours, chances are the bill would be accepted.
If you report 50 hours, 25% more than expected. Yes, they would question you. And you expect that. So, you would have to explain why you spend 25% more than what you agreed earlier.
If you did all that and the client still give you hard time by asking you to provide proof of time spent. You and your client are having the mutual trust issue. This is another problem, not what you're asking in this question.