Find two angles whose sum or difference is #345^@# and whose tangent we know (or can figure out).
The angle we know: #30^2, 45^@, 60^2# and multiples of these.
#345^@ - 30^@ = 315^@# is that a multiple of a special angle?
#315 = 270^@ + 45^@# so, yes, it is the quadrant IV, #45^@# angle.
So we can use:
#tan 345^@ = tan(30^@+315^@)#
#= (tan30^@+tan315^@)/(1-tan30^@tan315^@)#
#= (sqrt3/3+(-1))/(1-(sqrt3/3)(-1)) = (sqrt3/3-1)/(1+sqrt3/3))#
# = (sqrt3 -3)/(3+sqrt3) = (-3+sqrt3)/(3+sqrt3) (3-sqrt3)/(3-sqrt3) #
#=(-9+6sqrt3-3)/(9-3) = (-12+6sqrt3)/6 = -2+sqrt3#
