Why does one light bulb keep burning out on my low voltage landscape lights but the other 5 don't?
It is possible that there is a drop in voltage of the along the conductor through which the current is flowing. Landscape light bulbs can only burnout in case of voltage surge. You should use a voltmeter to try and locate the cause of the surge. For more tips visit this site http://www.ehow.com/how_6802231_diy_-troubleshoot-landscape-lights.html.
The sealant that protects the light from moisture is probably bad. Get a new seal or try some silicone, or there is a damaged wire right before or after that light. In that case electrical tape should work.