The easy answer first: if your system is running, then your CPU cooler is doing the most important part of its job. With no CPU cooling, the chip would overheat very quickly and the entire system would shut down and refuse to start up again until it cooled off.
Now, a better answer is whether it is doing the BEST job, not just "adequate". For this you need a tool to observe CPU temperature, and some knowledge of what is "normal" for your CPU.
In BIOS Setup where fan settings are configured there should be a display of the CPU temperature. This will tell you the current temperature, but it's hard to get that reading when you are doing real work. Instead, check the CD that came with your mobo for a monitoring utility able to show you temperatures, fan speeds, voltages, etc. It's a little application that runs under Windows, so you can set it running and watch things on your screen as you do real work. That gets you real-time temperature readings.
Now, what it "normal" for a CPU's internal temperature varies a lot from one type to another. Try searching the internet for information of "normal" and "maximum" temperatures for your particular CPU chip. Then you can compare to what you are getting. As long as you're within those ranges, your cooling system is doing exactly what it should.