Is using the book cheating?

For Daily Chess, no. For Live, using your memory of book, no, looking up the line, yes.

why wouldn't that be cheating? you are using a book to make the most accurate moves in the beginning. That takes away your opponents chances of capitalizing on your weaknesses and is absolutely cheating! You should study your mistakes after a game

It isn't cheating in Daily (which is correspondence chess). You can use books, videos, magazines, databases. You can't use engines or tablebases or other players.

For Live, you just get to use your brain .

https://support.chess.com/customer/en/portal/articles/1444879-fair-play-on-chess-com-what-you-need-to-know

What are the rules?

• No chess programs or engines (e.g. Chessmaster, Fritz, Komodo, Houdini, Stockfish, Chessbase with any active UCI engine, etc.) can be used to analyze positions in ongoing games at any time.  
• In Daily Chess (turn-based games with several days per move), you may consult books or databases (including the Chess.com Explorer) for opening moves. "Tablebases" - which are specialized databases of particular endgame positions - may NOT be used at any time. Further, you may not consult an engine to provide an opinion on your opening database, self-preparation or analysis that would relate to a particular game-in-progress on Chess.com.
• In Live Chess, no outside assistance OF ANY KIND is permitted.
• Fixing game results by playing with multiple accounts or losing intentionally is also against the rules.

Static resources like books, databases, Youtube videos, etc., are permitted in daily chess.  This might seem strange to people who are new to correspondence chess, but correspondence chess has historically had different rules from OTB tournaments.  Research in books, etc., has always been a part of correspondence chess, going back to the days it was played by snail mail.

In fact, in official correspondence tournaments run by ICCF and various national correspondence federations, they even allow engine use.  Chess.com is actually breaking from the traditional rules of correspondence chess by forbidding engine use in daily chess.

Here's an article on correspondence chess discussing some of the issues with engine use: https://www.bccachess.org/about/correspondence-chess-2/

Basically, daily chess is comparable to an open book exam where you have access to the textbook but still have to figure out how to apply the general principles the textbook teaches to the specific examples in the examination.

Live chess, on the other hand, is intended to roughly mimic OTB chess, so the rules forbid any form of assistance.

Thank you for that thorough explanation

I'll throw in my 2c and let you guys pick it (or pick on me about it )

I think the simplest reason for which the use of opening books and such is allowed in daily chess is because it's simply impossible to prevent it. How would chess.com know if a certain player is using an opening book or just his memory specifically for the 10 moves in the opening? Since the opening theory has grown to such an incredible scale, it's quite reasonable to think that half-decent players (i.e. the ones that are concerned about what to do in the opening and about using the theory developed so far) know such moves to some extent. How can a website "prove" that the 9th move you made in the opening is really your own (memorized or just discovered) or just picked from any opening theory source? Simply impossible.

That's not the case with the transition into the middlegame, where any reputable site would develop tools to detect use of outside help (engines, especially). However, if you fast forward 50 years into the future, you can see how this will start to affect the middlegame too, since more and more lines will be analyzed exhaustively and opening books will go farther and farther down the move order.

My advice? Just play Chess960 and play your own chess right from the opening, not the middlegame. And you can be pretty sure that your opponent is trying to solve the same problem you have in the opening, instead of relying on already proven lines. I find it totally disheartening to have to play "by the book" for the first 10+ moves, just so you can get to the middlegame with a fair chance, instead of everyone developing their own game plan right from the get go.

The inability to police it is a factor, although if an 800-rated player regularly follows random obscure lines 15-20 moves deep before hanging a rook as soon as he leaves book, you know exactly what's going on.  But a big factor is the heritage of correspondence chess.

Correspondence chess grew up with the idea of taking your time and doing your research to play as strong a game as you possibly can.  When the first chess computers started to come along, they naturally fit into that; they were weak, but could provide an extra check for any tactics you might miss; still, a strong player could get by without them.  Over time the computers grew stronger and came to dominate correspondence chess; now, the best correspondence players are the ones who do very extensive opening research, know the strengths and weaknesses of engines, and know how to handle positions that engines evaluate poorly; it's not all about having the best hardware.

Chess.com and other sites that offer correspondence chess are partly playing to this tradition, but their main target audience is the amateur player who has nothing to contribute to an engine in centaur chess, and just wants to play a game without time pressure in a manner that fits his schedule.  Thus, allowing static resources such as books and databases but forbidding engine use makes perfect sense.

Which book exactly?

"Book" moves are ones that are known opening moves.  Nobody plays an original move as their first move.  The position after 1.e4 e5 2.Nf3 Nc6 has been analysed extensively, with some lines analysed to a depth of 10, 15, 20 moves or more, and if you look at a database of master-level games, you can see these lines have been played dozens, hundreds, or even thousands of times.  These moves are all considered "book" moves, and you can find books dedicated to particular openings that will analyse them in-depth and provide suggested plans for the resulting middlegame positions.

At some point, however, unless you copy a master-level game from beginning to end, someone is going to play an original move.  At that point you're said to have "left book".