You can not authenticate a recording with a phone bill. Either party to the conversation can authenticate it. As for deposition costs you pay by the page not by the hour so playing all or part of the recording won't effect the cost. Assuming you received a discovery demand for statements have you served a copy of the recording on your adversary?
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I would definitely consult with an attorney on this, because your questions raises a number of possible objections to this proposed evidence and/or things which must be established. I take it these are recordings of both telephone calls and surreptitious audio recordings of conversations with a witness or opposing party.
One issue is consent, and reasonable expectation of privacy, but I believe NY is a "one party" state where only one party (you) needs to consent to a recording without announcing the conversation is being recorded. Another issue is "chain of custody": how these recordings came into being and that they have not been edited or worse, altered (spliced together, edited or "photoshopped" with a tool like Pro Tools).
One way of side-stepping this problem for purposes of both the deposition and the trial might be to use the tapes as impeachment rather than direct evidence. In other words, you would ask the witness a question like "Do you recall on or about May 1, 2012, we had a conversation about you taking my tuna fish sandwich with my name written on it from the break room refrigerator and eating it?". If she said "yes", you might follow up, "did you admit to eating the sandwich" and hopefully get the desired "Yes, I ate it" answer. But if she either denied remembering the conversation, or her admitting eating the tuna fish sandwich, I might offer the audiotape as evidence (after laying a foundation), playing it and asking "does this recording refresh your recollection" and "after hearing this recording, do you now want to change your answer to my previous question".
If you still get resistance, or objections, you can try to introduce the tape as evidence by qualifying how it came into being and the chain of custody, etc. The difference here is "direct evidence" vs. "evidence used for impeachment", but in either case, people tend to believe recordings, videos, documents and other physical evidence over oral testimony so the practical effect might be the same.
If you use this tape, be prepared to hand over the original of the tape to the other side for inspection and copying to prove it has not been altered or there is not other evidence on the tape that supports contentions of the other side.
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You really need to have an attorney do this. Your question is not as specific as you think, and the problems that go along with your wanted evidence are far more pervasive than you know.
An attempt to "keep deposition costs down" is really a penny wise pound foolish approach. If the recording is as good as you think, hire counsel and get the project done right. That is the way to get the best value here.
You simply have the person who made the recording take the stand and explain the five "W's" as the recording's being made. I don't see how an objection to its admisibility would be sustained in a state court or a federal court.