Why paper money and coins use together?

I assume you're asking why we have both Coins and Paper Money. The simple answer is that coins are much easier to use and more durable, especially in small denominations because those amounts are used much more often.

For example, most transactions involve a few pennies here and there. If there were a one-cent bill you'd be carrying dozens of them after a while, and they'd wear out very quickly from all the handling. The government would have to print tens of billions of 1-cent bills every year just to keep up with demand. 1-cent coins, however, last for 40 or 50 years on average so it's a lot easier and cheaper to make them from metal.

When you get to higher denominations, it's more cost-effective to print bills because they don't get used as often. That's why $50 bills sometimes circulate for 30 or more years before they wear out.

The place where things become interesting is when you're dealing with low-denomination bills. $1 bills are needed A LOT in change-making, so they wear out very quickly - anywhere from 12 to 18 months on average. It then becomes expensive to print all the $1 bills needed which is why many people are calling for them to be replaced with a coin that would last 25 or 30 years. Most countries have already done that; in fact, Canada has $2 coins and the EU has €2 coins. For a number of reasons, political groups in the U.S. have lobbied in favor of keeping the $1 bill even though the government estimates it would save about $500 million to $1 billion a year if we used a $1 coin instead.