We use the fact that 345° = 315°+30°, two special angles, and the identity

,

From our knowledge of special angles:

tan(315°) = -1 and tan(30°) =  = 

tan(345°) = tan(315°+30°) =   =  = 

To simplify that, first we multiply top and bottom by 3 and distribute
to clear of fractions in the numerator and denominator:

 = 

Now we rationalize the denominator by multiplying top and bottom by 3-√3,
multiplying out and simplifying:

 =  =  =  =  =

 =  =  = -2+√3

It's always a good idea to check such answers with a calculator,
getting and comparing the decimal approximations:

By calculator tan(345°) = -.2679491924

and by calculator

-2+√3 = -.2679491924

Edwin