I drive a 1999 Lexus RX300.  I am towing a U-Haul trailer. After dark
I was waved over by a motorist to tell me the tail lights were out.
They were out on the Lexus, and no lights on the trailer were on.  I
need to know 1) Where is the tail light fuse on the RX300?  The Lexus
manual was not helpful.  There is a fuse compartment under the hood
but nothing was labeled "TAIL".  The manuals says there is another
fuse box "on the driver's side instrument cluster" but I could not
find it.  A diagram or photograph would help.  2) I need to know if
when towing using this vehicle, the 10 amp tail lamp fuse should be
replaced with a higher amp fuse, and if so, what amperage?  This car
has the 3500 pound towing package.

Have you checked all the fuses to see if one is blown. Have you called
a dealer, maybe they would be willing to tell you. I would assume that
the stock fuse amps would be correct. There's probably a short in the
U-haul trailer that blew the fuse if the lights worked before.

The Lexus dealer said that one of two relays had burned out, at a
minimum cost of $500 for one relay alone, plus labor.  I took the car
to an independent shop that specializes in foreign cars.  They
identified the broken relay, and offered to replace it with a new one
for $400 for parts and labor, or to repair the broken one for $180. 
I'm trying the latter option.

I did check all fuses to see if one is blown.  I learned that the
RX300 has multiple fuse boxes, one under the hood, and one inside,
among others.  The one inside is located behind the coin holder, which
is located at the driver's left knee.  To expose the fuse panel, open
the coin drawer, then pull it up and out.  The owner's manual is
useless on this detail, and I could not find this needed information
despite extensive Googling.  The tail light fuse had NOT blown.

So the fuse was good, the $500 relay was bad, and we can consider this
question answered.   Any Google Answerers should be forgiven for not
finding out this information, as it probably does not appear anywhere
online.

Thanks, folks....

/rich