You *can* hide the user agent string but whether is is of much benefit to do so from a security standpoint is deebateable. You can determine the browser a person is using (within reasonable estimates if no always exactly) by other means. WHat langauges does it support, does it accept activeeX, does it sue javascript (if it does it will reveal a LOT more info), does it accept plugins, which ones are isntalled, etc.
There is a website that does a browser fingerprinting demo that can be a real eye opener.
https://panopticlick.eff.org/
For more info on browser fingerprinting just do a web search for the term.
You also may face problems on those websites that are still mired in the 90's that do browser sniffing to display targeted content to different browsers.
Whether you feel it is worth while is up to you. For myself I don't bother too much.