It has to be simple.
The high beam and low beam share a ground wire, so if the high beam works there has to be a good ground for low beam.
The high beam and low beam share a common power source, switched by the head light selector switch. If the high beam works, there must be power coming onto the selector switch.
All that leaves is the Red/Yellow wire that goes from the selector switch to the low beam terminal on the headlight connector.
Using a $0-$10 Volt/Ohm Meter (you can get them for free sometimes with a coupon at Horrid Fright*, but can usually buy one for $10 at Home Depot), start tracing the circuit. Unplug the handlebar switch connector.
using the "Ohms" setting, check for a good circuit (0ish ohms) between the Brown wire and the Red/Black wire with the switch in the 'Hi Beam' position. This is done by touching the meter's probes to the appropriate pins in the connector. That circuit is known to be good, so you should see connectivity (0ish Ohms). Switch to 'Lo Beam and check the connectivity of the Brown and the Red/Yellow wire. If that is not good you have found a problem. If it is good, proceed.
Check the connectivity of the Red/Yellow wire from the connector that plugs into the switch to the headlight connector. If that is good, check the low beam filament in the headlamp.
If it still all check good, plug it back together and see if it works. The contacts in the connectors can become corroded and give problems; unplugging them and re-plugging them can cure that. A bit of Vaseline or dielectric grease can help seal the connections from moisture and dust.
In case you don't have an electrical diagram, this post has one you can use: http://www.klrforum.com/165769-post1.html
Let us know what you find.
*Horrid Fright=Harbor Freight. Sort of like your TradeTools, except everything is half the price, comes from China, and smells funny.